---
abstract: |
    Work in ${\mathsf{ZF}}$ or ${\mathsf{ZF}}_2$ (second order ${\mathsf{ZF}}$), as appropriate. Recall that a Reinhardt cardinal is the critical point of a (non-trivial) elementary embedding $j:V\to V$. Beyond these, one has super-Reinhardt, total Reinhardt and Berkeley cardinals. We prove the following results.

    Let $X$ be a set and $A$ a class. Then (i) if there is a Reinhardt cardinal then $V\neq{\mathrm{HOD}}(X)$, and (ii) if $V$ is total Reinhardt or there is a Berkeley cardinal then $V\neq{\mathrm{HOD}}_A(X)$.

    Let $\delta$ be a limit ordinal and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. Then (i) $j$ is not definable from parameters over $V_\delta$, and (ii) there is $n<{\omega}$ such that the $n^{\mathrm{th}}$ iterate $j^n=j(j(\ldots (j))):V_\delta\to V_\delta$ is fully elementary; in fact, $j^n:(V_\delta,A)\to(V_\delta,j^n(A))$ is fully elementary for all $A{\subseteq}V_\delta$.

    Let $\delta$ be any ordinal and $j:V_{\delta+1}\to V_{\delta+1}$ be fully elementary. Then $j$ is not definable over $V_{\delta+1}$ from parameters in $V_\delta$.

    Suppose $V=L(V_\delta)$ and $\operatorname{cof}(\delta)>{\omega}$. Then there is no $\Sigma_1$-elementary $j:V_\delta\to V_\delta$.

    Let $G$ be $(V,{\mathbb P})$-generic for some ${\mathbb P}\in V$. Then (i) if $V[G]$ has a super-Reinhardt cardinal, then $V$ has a super-Reinhardt cardinal; (ii) if ${\mathbb P}\in V_\delta$ and $V[G]{\models}$“$V_\delta^{V[G]}$ is total Reinhardt” then $V{\models}$“$V_\delta$ is total Reinhardt”; and (iii) if $V[G]$ has a set of ordinals which is not in $V$, then $V[G]$ has no elementary $j:V[G]\to M{\subseteq}V$.

    We also develop the theory of ultrapowers by extenders under ${\mathsf{ZF}}$, and show that if there is a proper class of Löwenheim-Skolem cardinals, then being the critical point of an elementary $j:V\to M$ (with $M$ transitive) is first-order.
author:
- |
    Farmer Schlutzenberg[^1]\
    farmer.schlutzenberg@gmail.com
bibliography:
- '../bibliography/bibliography.bib'
title: |
    Reinhardt cardinals and non-definability\
    (draft 1)
---

This draft is hopefully reasonably complete, but is presently somewhat repetitive and will be better organized in a later version.

Introduction
============

A *Reinhardt cardinal*, introduced by William Reinhardt in [@reinhardt_diss] and [@reinhardt_remarks], is the critical point of an elementary embedding $j:V\to V$. Kunen showed in [@kunen_no_R] that if $V{\models}{\mathsf{ZFC}}$ then there is no such $j$. Thus, Reinhardt embeddings (that is, $j:V\to V$) are considered with background theory ${\mathsf{ZF}}$, possibly augmented with fragments of ${\mathsf{AC}}$.

We consider here Reinhardt cardinals and embeddings, and variants thereof, such as elementary embeddings $j:V_\delta\to V_\delta$, and also stronger large cardinal notions introduced by Woodin, which we (partially) describe below; these are taken from the Bagaria/Koellner/Woodin paper [@woodin_koellner_bagaria], which discusses other aspects and in which the reader may also be interested.

The main focus of the paper is on proving that under certain simplicity assumptions, there can be no Reinhardt cardinal (or, possibly, a variant thereof); analogously, there can be no elementary $j:V_\delta\to 
V_\delta$ which is too simple (maybe with further assumptions). We also discuss some other related topics. These include a development of ultrapowers and extenders under ${\mathsf{ZF}}$, and discussion of Reinhardt and other embeddings appearing in set-generic extensions.

Other related works include Apter/Sargsyan [@sarg_apt_jonsson_V_to_V], which deduces certain partition properties from (enough) Reinhardt cardinals, Cutolo [@cutolo_cofinality], [@cutolo_structure] on Berkeley cardinals, and Gabriel Goldberg has unpublished results regarding the consistency strength of Reinhardt cardinals in comparison with large cardinals compatible with ${\mathsf{AC}}$. (However, Goldberg’s work was not familiar to the author at the time of writing.) Other than for the definition of *Berkeley cardinal*, this paper is independent of these other works. The forcing arguments here relate to Laver [@laver_vlc].

We now list some basic terminology and definitions, which are pretty standard. Note however, that the basic background theory is ${\mathsf{ZF}}$ (in fact, many of the results require much less than ${\mathsf{ZF}}$). In this context, certain definitions from the ${\mathsf{ZFC}}$ context need to be modified appropriately.

The *language of set theory with predicate and embedding* ${\mathcal{L}}_{\dot{\in},\dot{A},\dot{j}}$ is the first order language with binary predicate symbol $\dot{\in}$, predicate symbol $\dot{A}$ and function symbol $\dot{j}$.

The theory *${\mathrm{ZFR}}_{\dot{A}}$* is the theory in ${\mathcal{L}}_{\dot{\in},\dot{A},\dot{j}}$ with all ${\mathsf{ZF}}$ axioms (with $\dot{\in}$ being membership), allowing all formulas of ${\mathcal{L}}_{\dot{\in},\dot{A},\dot{j}}$ for the Separation and Collection schemes, together with the (single) axiom asserting that $$\text{``}\dot{j}:(V,\dot{A},\dot{\in})\to(V,\dot{A},\dot{\in})\text{ is 
}\Sigma_1\text{-elementary.''}\footnote{One could
add the axiom scheme, where for each 
$n\in{\mathbb{N}}$, we have the assertion that $\dot{j}$
is $\Sigma_n$-elementary, but we will see that
this already follows.}$$ And ${\mathrm{ZFR}}$ is the theory ${\mathrm{ZFR}}_{\dot{A}}+$“$\dot{A}=\emptyset$”. We will often drop the “dot” from the notation, just writing ${\mathrm{ZFR}}_A$, etc, where this is clear.

Let $j:V_\delta\to V_\delta$ or $j:V\to V$ be $\Sigma_1$-elementary and $\kappa_0={\mathrm{cr}}(j)$ and $\kappa_{n+1}=j(\kappa_n)$. Let $\lambda=\sup_{n<{\omega}}\kappa_n$. (Note that $\lambda\leq\delta$.) We write $\kappa_{n,j}=\kappa_n$ and $\lambda_j=\kappa_{{\omega},j}=\lambda$.

Given a structure $M=({{\left\lfloor}M{\right\rfloor}},\in^M,\ldots)$ with universe ${{\left\lfloor}M{\right\rfloor}}$, and given $A{\subseteq}{{\left\lfloor}M{\right\rfloor}}$, we say that $A$ is *amenable* to $M$ iff for each $x\in{{\left\lfloor}M{\right\rfloor}}$, we have $$\{y\in{{\left\lfloor}M{\right\rfloor}}\bigm|y\in A\text{ and }y\in^M x\}\in{{\left\lfloor}M{\right\rfloor}}.$$

A *class* of a model $M$ of ${\mathsf{ZF}}$ is a collection $X{\subseteq}M$ such that $X$ is amenable to $M$ and $(M,X){\models}{\mathsf{ZF}}$. (Here we must be working in some background model which sees $M$ and collections $X{\subseteq}M$, where we make this definition.)

Second order set theory is denoted ${\mathsf{ZF}}_2$. Models of this theory are of the form $(V,\dot{\in},P)$, where $(V,\dot{\in}){\models}{\mathsf{ZF}}$ and $P$ is a collection of classes of $V$, satisfying the ${\mathsf{ZF}}_2$ axioms. Whenever we work in ${\mathsf{ZF}}_2$, when we speak of “classes” we mean in the formal sense specified by ${\mathsf{ZF}}_2$; or when we work in a structure $(V,\dot{\in},P){\models}{\mathsf{ZF}}_2$, we mean one of the elements of $P$. Given a transitive set $W$ and $P{\subseteq}{\mathcal{P}}(W)$, we write $(W,P){\models}{\mathsf{ZF}}_2$ iff $(W,\in,P){\models}{\mathsf{ZF}}_2$.

Assume ${\mathsf{ZF}}$ (or less). Let $\delta\in{\mathrm{OR}}$ be a limit (ordinal). Then the *cofinality* of $\delta$ is the least ordinal $\gamma$ such that there is $f:\gamma\to\delta$ which is cofinal in $\delta$. We say that $\delta$ is *regular* iff $\operatorname{cof}(\delta)=\delta$. We say that $\delta$ is *singular* if it is not regular. We say that $V_\delta$ (or just $\delta$) is *inaccessible* iff there is no $(\gamma,f)$ such that $\gamma<\delta$ and $f:V_\alpha\to\delta$ is cofinal in $\delta$.

Assume ${\mathsf{ZF}}$ and let $\delta\in{\mathrm{OR}}$ be a limit. It is easy to see that (i) $\delta$ is inaccessible iff (ii) $(V_\delta,X){\models}{\mathsf{ZF}}$ for every $X{\subseteq}V_\delta$ iff (iii) for every $X{\subseteq}V_\delta$ there is a club of $\alpha<\delta$ such that $(V_\alpha,X{\cap}V_\alpha){\preccurlyeq}(V_\delta,X)$.

Assume ${\mathsf{ZF}}$. Given $\delta\in{\mathrm{OR}}$, note that $V_\delta$ is inaccessible iff $(V_\delta,V_{\delta+1}){\models}{\mathsf{ZF}}_2$.

The following notions are taken from [@woodin_koellner_bagaria]:

Work in ${\mathsf{ZF}}_2$.

We say that $\kappa\in{\mathrm{OR}}$ is *super-Reinhardt* iff for every $\lambda\in{\mathrm{OR}}$ there is a class $j$ such that $(V,j){\models}{\mathrm{ZFR}}$ and ${\mathrm{cr}}(j)=\kappa$ and $j(\kappa)\geq\lambda$.

Given a class $A$, we say that $\kappa\in{\mathrm{OR}}$ is *$(A,{<{\mathrm{OR}}})$-Reinhardt-reflecting* iff for every $\lambda\in{\mathrm{OR}}$ there is a class $j$ such that $(V,A,j){\models}{\mathrm{ZFR}}_A$ and ${\mathrm{cr}}(j)=\kappa$ and $j(\kappa)\geq\lambda$. We say that $V$ is *totally Reinhardt* iff for every class $A$ there is $\kappa\in{\mathrm{OR}}$ which is $(A,{<{\mathrm{OR}}})$-Reinhardt-reflecting.

Given a limit ordinal $\delta$ and an elementary $j:V_\delta\to V_\delta$, and given $C{\subseteq}V_\delta$, we define $$j(C)=\bigcup_{\alpha<\delta}j(C{\cap}V_\alpha).$$ Similarly in the case that $(V,A,j){\models}{\mathrm{ZFR}}_{\mathrm{A}}$ and $C$ is a class of $(V,A,j)$.

Work in ${\mathsf{ZF}}$ (or less). For $\kappa<\delta\in{\mathrm{OR}}$, we say that $\kappa$ is *$V_\delta$-super-Reinhardt* iff $(V_\delta,V_{\delta+1}){\models}$“$\kappa$ is super-Reinhardt”. We say that *$V_\delta$ is totally Reinhardt* iff $(V_\delta,V_{\delta+1}){\models}$“$V_\delta$ is totally Reinhardt” (hence, $V_\delta$ is inaccessible). For *Berkeley cardinal*, see [@woodin_koellner_bagaria].

Work in ${\mathsf{ZF}}$. Let $C$ be a class. Then ${\mathrm{OD}}_C(X)$ denotes the class of all sets $y$ such that $y$ is definable from the predicate $C$ and parameters in ${\mathrm{OR}}\cup X$. And ${\mathrm{HOD}}_C(X)$ denotes the class of all $y$ such that ${\mathrm{trcl}}(\{y\}){\subseteq}{\mathrm{OD}}_C(X)$. And ${\mathrm{OD}}(X)={\mathrm{OD}}_\emptyset(X)$ and ${\mathrm{HOD}}(X)={\mathrm{HOD}}_\emptyset(X)$.

In §§\[sec:no\_Reinhardt\_in\_L(X)\], \[sec:no\_Reinhardt\_in\_HOD(X)\], \[sec:Berkeley\] we will prove various results indicating that if there are Reinhardt cardinals or beyond, then the universe is complex:

Assume ${\mathsf{ZF}}_2$. Let $X$ be a set and $A$ a class. Then:

1.  \[item:V=HOD(X)\_no\_R\] (\[tm:V=HOD(X)\_no\_R\]) If $V={\mathrm{HOD}}(X)$ then there is no Reinhardt cardinal.

2.  \[item:V=L(X)\_no\_R\] (\[tm:V=L(X)\_no\_R\]) (Hence if $V=L(X)$ then there is no Reinhardt cardinal.)

3.  (\[cor:V=HOD\_A(X)\_no\_tR\]) If $V={\mathrm{HOD}}_A(X)$ then $V$ is not total Reinhardt.

4.  (\[tm:V=HOD\_A(X)\_no\_B\]) If $V={\mathrm{HOD}}_A(X)$ then there is no Berkeley cardinal.

In fact, in parts \[item:V=HOD(X)\_no\_R\] and \[item:V=L(X)\_no\_R\] we don’t need all of ${\mathsf{ZF}}_2$; the real theorem is:

${\mathrm{ZFR}}$ proves that there is no set $X$ such that $V={\mathrm{HOD}}(X)$.

We will also use some of the methods involved in the proof of these results to exhibit some interesting forcing extensions which follow from the existence of Reinhardt embeddings:

Let $(M,j){\models}{\mathrm{ZFR}}$ and $\kappa={\mathrm{cr}}(j)$. Then for all $\alpha\in{\mathrm{OR}}^M$ there is $\eta\geq\alpha$ and a set-forcing ${\mathbb P}\in M$ which forces that in $M^{\mathbb P}$,

1.  $V_\kappa^{M^{\mathbb P}}=V_\kappa^M$,

2.  $M^{\mathbb P}{\models}$“There is an elementary $k:V_\eta\to V_\eta$ with $\kappa_{{\omega},k}=\kappa$”.

Also, for all $\alpha\in{\mathrm{OR}}$ there is $\eta\geq\alpha$ and $\bar{\lambda}<\kappa$ with $M{\models}$“$\bar{\lambda}$ is inaccessible”, and a set-forcing ${\mathbb P}\in M$ which forces that in $M^{\mathbb P}$,

1.  $V_{\bar{\lambda}}^{M^{\mathbb P}}=V_{\bar{\lambda}}^M$,

2.  $M^{\mathbb P}{\models}$“There is an elementary $k:V_\eta\to V_\eta$ with $\kappa_{\omega}(k)={\bar{\lambda}}$”.

3.  $(\bar{\lambda}^+)^{M^{\mathbb P}}=\kappa$.

Let ${\mathcal{L}}_{\dot{A}}$ be the language of set theory augmented with a one-place predicate symbol $\dot{A}$; given a structure of the form $M=(U,\in^M,A)$, where $A{\subseteq}U$, we interpret $\dot{A}$ as $A$, and write $A^M=A$. This generalizes in the obvious manner to the language ${\mathcal{L}}_{\dot{A}_1,\ldots,\dot{A}_n}$ and structures $M=(U,\in^M,A_1,\ldots,A_n)$.

Let $M=(U,\in^M,A_1,\ldots,A_n)$ be a first-order structure with universe $U$ and $A{\subseteq}U$. (We normally abbreviate this by just writing $A{\subseteq}M$.) We say that $A$ is *definable over* $M$ *from parameters* if there is a first-order formula $\varphi\in{\mathcal{L}}_{\dot{A}_1,\ldots,\dot{A}_n}$ and some $\vec{x}\in M^{<{\omega}}$ such that for all $y\in M$, we have $y\in A$ iff $M{\models}\varphi(\vec{x},y)$. This is naturally refined by *$\Sigma_n$-definable from parameters*, if we require $\varphi$ to be $\Sigma_n$, and by *from parameters in* $X$, if we restrict to $\vec{x}\in X^{<{\omega}}$ (where $X{\subseteq}M$), and by *from $\vec{x}$*, if we may use only $\vec{x}$.

In §\[sec:j:V\_delta\_to\_V\_delta\], we rule out certain cases of too simple elementary $j:V_\delta\to V_\delta$. Suzuki proved (see Fact \[fact:suzuki\_no\_def\_j\] for the proof):

If $(V,j){\models}{\mathrm{ZFR}}$ then $j$ is not definable over $V$ from parameters.

We will prove certain variants of this result:

Assume ${\mathsf{ZF}}$ and let $\delta\in{\mathrm{OR}}$ be a limit and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. Then $j$ is not definable from parameters over $V_\delta$.

Assume ${\mathsf{ZF}}$ and let $\delta\in{\mathrm{OR}}$ and $j:V_{\delta+1}\to V_{\delta+1}$ be $\Sigma_1$-elementary. Then $j$ is not definable over $V_{\delta+1}$ from parameters in $V_\delta$.

However, if $j:V_{\delta+1}\to V_{\delta+1}$ is $\Sigma_1$-elementary where $\delta$ is a limit, then a standard observation shows that $j$ *is* definable over $V_{\delta+1}$ from the parameter $j{\!\upharpoonright\!}V_\delta$: Given $X{\subseteq}V_\delta$, we have $j(X)=\bigcup_{\alpha<\delta}j(X{\cap}V_\alpha)$.

Recall that:

Given a set $X$, a *prewellorder* on $X$ is a reflexive linear order $\leq$ on $X$ such that the strict part $<$ of $\leq$ is wellfounded.

So the ordertype of any prewellorder is an ordinal. It is easy to see that prewellorders on $X$ of ordertype $\beta\in{\mathrm{OR}}$ and surjections $\pi:X\to\beta$ are essentially the same thing.

\[lem:reg\_non\_inacc\] Let $\delta$ be regular but not inaccessible. Let $\alpha$ be least such that there is a cofinal map $\pi:V_\alpha\to\delta$. Then (i) $\alpha=\beta+1$ for some $\beta$, and (ii) for every cofinal function $\pi:V_\alpha\to\delta$, ${\mathrm{rg}}(\pi)$ has ordertype $\delta$.

Suppose (i) fails, so $\alpha<\delta$ is a limit. By minimality of $\alpha$, we can define a function $f:\alpha\to\delta$ by $f(\beta)=\sup\pi``V_\beta$. Since $\pi$ is cofinal, so is $f$, but then $\delta$ is singular, a contradiction. Part (ii) is an immediate consequence of regularity.

\[dfn:codcof\] Assume ${\mathsf{ZF}}$ and let $\delta\in{\mathrm{OR}}$ be regular but not inaccessible. Let $\alpha+1$ be the least ordinal such that there is a cofinal $\pi:V_{\alpha+1}\to\delta$ (see Lemma \[lem:reg\_non\_inacc\]). The *coded cofinality* ${\mathrm{codcof}}(\delta)$ of $\delta$ is the set $P$ of all prewellorders of $V_{\alpha+1}$ of ordertype $\delta$ (which is non-empty, by Lemma \[lem:reg\_non\_inacc\]).

The proof of Theorem \[tm:no\_def\_Sigma\_1\_V\_delta\_to\_V\_delta\] relies on the following theorem; it is a variant of the fact that whenever $M,N{\models}{\mathsf{ZF}}$ and $j:M\to N$ is $\Sigma_1$-elementary and $\in$-cofinal, then $j$ is fully elementary (see [@kanamori Proposition 5.1], where the stated proof literally assumes that $M,N$ are transitive proper class inner models, or see Fact \[fact:ZF\_Sigma\_1-elem\_cofinal\_implies\_full\_elem\]).[^2]

Assume ${\mathsf{ZF}}$ and let $\delta\in{\mathrm{OR}}$ be a limit and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. Assume that either:

1.  $V_\delta$ is inaccessible, or

2.  $\delta$ is regular and $j({\mathrm{codcof}}(\delta))={\mathrm{codcof}}(\delta)$, or

3.  $\delta$ is singular and $j(\operatorname{cof}(\delta))=\operatorname{cof}(\delta)$.

Then $j$ is fully elementary, and in fact, for every $A{\subseteq}V_\delta$, $$j:(V_\delta,A)\to(V_\delta,j(A))$$ is fully elementary with respect to the language ${\mathcal{L}}_{\dot{A}}$.

We will also observe that the assumptions above regarding fixed points of $j$ are not too much to ask for.

Assume ${\mathsf{ZF}}$ and let $\delta\leq{\mathrm{OR}}$ be a limit and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. Then $j^1=j$ and $j^{n+1}=j(j^n)$ for $n<{\omega}$.

Assume ${\mathsf{ZF}}$ and let $\delta\in{\mathrm{OR}}$ be a limit and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. Then:

1.  for each $n<{\omega}$, $j^n:V_\delta\to V_\delta$ is $\Sigma_1$-elementary,

2.  if $x\in V_\delta$ and $j^n(x)=x$ then $j^m(x)=x$ for all $m\in[n,{\omega})$,

3.  if $\gamma<\delta$ then there is $n<{\omega}$ such that $j^n(\gamma)=\gamma$.

4.  if $\delta$ is regular but not inaccessible then there is $n<{\omega}$ such that $j^n({\mathrm{codcof}}(\delta))={\mathrm{codcof}}(\delta)$.

Going beyond definability from parameters over $V_\delta$, it is natural to ask whether an elementary $j:V_\delta\to V_\delta$ can appear in $L(V_\delta)$. This question can also be refined by looking at the hierarchy of $L(V_\delta)$.

Recall that for a transitive set $X$, ${\mathcal{J}}(X)$ is one step in Jensen’s ${\mathcal{J}}$-hierarchy above $X$; that is, it is the rudimentary closure of $X\cup\{X\}$. One key fact is that $${\mathcal{J}}(X){\cap}{\mathcal{P}}(X)=\{A{\subseteq}X\bigm|A\text{ is definable from parameters over 
}X\}.$$ The ${\mathcal{J}}$-hierarchy above $X$ is then given by setting ${\mathcal{J}}_{\xi+1}(X)={\mathcal{J}}({\mathcal{J}}_\xi(X))$, and by taking unions at limits $\xi$.

So Theorem \[tm:no\_def\_Sigma\_1\_V\_delta\_to\_V\_delta\] just asserts that if $\delta\in{\mathrm{OR}}$ is a limit then there is no $\Sigma_1$-elementary $j:V_\delta\to V_\delta$ which is in ${\mathcal{J}}(V_\delta)$. In fact we will show:

(\*\*\*This is currently formally a conjecture, awaiting writeup of some details...) Assume ${\mathsf{ZF}}$ and let $\delta\in{\mathrm{OR}}$ be a limit. Suppose $j:V_\delta\to V_\delta$ is $\Sigma_1$-elementary. Let $\lambda=\lambda_j$. Then $j\notin{\mathcal{J}}_\lambda(V_\delta)$.

We are not sure how far this goes in general. However, if the theory ${\mathrm{ZFR}}+$“$V_\lambda{\models}V={\mathrm{HOD}}$, where $\lambda=\lambda_j$” is consistent, then ${\mathsf{ZFC}}+V=L(V_\delta)+$“There is a fully elemetary $j:V_\delta\to 
V_\delta$” is also consistent (see \[rem:j\_in\_L(V\_delta)\]). But by adding the assumption that $\operatorname{cof}(\delta)>{\omega}$, we can rule this out:

Assume ${\mathsf{ZF}}$ and $V=L(V_\delta)$ where $\delta\in{\mathrm{OR}}$ is a limit with $\operatorname{cof}(\delta)>{\omega}$. Then there is no $\Sigma_1$-elementary $j:V_\delta\to V_\delta$.

In §\[sec:small\_forcing\], we will also prove that certain large cardinals cannot be “added” by small forcing:

Let $(V,P){\models}{\mathsf{ZF}}_2$. Let ${\mathbb Q}\in V$ be a poset and $G$ be $(V,{\mathbb Q})$-generic. Let $\kappa\in{\mathrm{OR}}$ with ${\mathbb Q}\in V_\kappa$. Suppose that $(V[G],P[G]){\models}$“$\kappa$ is super-Reinhardt”. Then $(V,P){\models}$“$\kappa$ is super-Reinhardt”, and moreover for every elementary $j:V[G]\to V[G]$ in $P[G]$, the restriction $j'=j{\!\upharpoonright\!}V$ is such that $j':V\to V$ is elementary and $j'\in P$.

Let $(V,P){\models}{\mathsf{ZF}}_2$ and $G$ be $(V,{\mathbb Q})$-generic for ${\mathbb Q}\in V$. Then:

1.  If $(V[G],P[G]){\models}$“There is a super-Reinhardt” then $(V,P){\models}$“There is a super-Reinhardt”.

2.  If $(V[G],P[G]){\models}$“$V[G]$ is total Reinhardt” then $(V,P){\models}$“$V$ is total Reinhardt”.

Moreover, these things are witnessed by restrictions of embeddings in $P[G]$.

Let $V{\subseteq}V[G]$ be a set-forcing extension via ${\mathbb P}{\subseteq}V_\eta^V$. Suppose that:

1.  $(V[G],j){\models}{\mathrm{ZFR}}$,

2.  $V[G]{\models}$“$V_\eta$ is wellordered and $\gamma^+$-${\mathsf{DC}}$ holds where $\operatorname{card}(V_\eta)=\gamma$”.

Then $j$ is amenable to $V$ and $(V,j{\!\upharpoonright\!}V){\models}{\mathrm{ZFR}}$.

Woodin showed that under ${\mathsf{ZFC}}$, there is no set-generic extension $V[G]$ of $V$ with an elemenary $j:V[G]\to V$. In §\[sec:generic\_j:V\[G\]\_to\_M\], we generalize this as follows:

Assume $(V,P){\models}{\mathsf{ZF}}_2$. Let $G$ be $(V,{\mathbb P})$-generic for some ${\mathbb P}\in V$, with $G\notin V$. Suppose there is $j\in P[G]$ with $j:V[G]\to M$ elementary and $M{\subseteq}V$. Then for every $\alpha\in{\mathrm{OR}}$, we have $${\mathcal{P}}(\alpha){\cap}V[G]{\subseteq}V.$$

In §\[sec:ultrapowers\_in\_ZF\] we also develop the theory of ultrapowers by extenders under ${\mathsf{ZF}}$, and in §\[sec:crit\_def\], show that under ${\mathsf{ZF}}+$“There is a proper class of Löwnenheim-Skolem cardinals”, the class of critical points of embeddings $j:V\to M$ (with transitive $M$) is definable.

These results leave various questions open, for example:

1.  Is the theory ${\mathrm{ZFR}}+$“$V={\mathrm{HOD}}_A(X)$ with a class $A$ and set $X$” consistent? In particular, with $A=j$? In particular, with $X=V_{\lambda+1}$ where $\lambda=\kappa_{\omega}(j)$? (Certainly $V\neq{\mathrm{HOD}}_j$, since ${\mathrm{HOD}}_j{\models}{\mathsf{AC}}$.)

2.  Is ${\mathrm{ZFR}}+$“${\mathsf{AC}}$ is forceable” consistent?

3.  Can one force a new Reinhardt cardinal or a new Berkeley cardinal with small forcing?

4.  Assume ${\mathsf{ZF}}$ and $V=L(V_\delta)$. Can there be a $j$ such that $(V_\delta,j){\models}{\mathrm{ZFR}}$? By Theorem \[tm:V=L(V\_delta)\_cof(delta)&gt;om\], this would imply $\operatorname{cof}(\delta)={\omega}$. (In the example in Remark \[rem:j\_in\_L(V\_delta)\], even if its hypothesis is consistent, we get $\delta=\lambda_j$, so we don’t have a model of ${\mathrm{ZFR}}$ there).

5.  Can one improve the fact of “no $j:V_\delta\to V_\delta$ in ${\mathcal{J}}_\lambda(V_\delta)$ (for $\delta$ a limit)”? In particular, if ${\mathcal{J}}_\alpha(V_\delta){\models}$“$V_\delta$ is inaccessible”, (but in $L(V_\delta)$, we have $\operatorname{cof}(\delta)={\omega}$)?

6.  What about if we replace $L(X)$ with $K(X)$, or $L(V_\delta)$ with $K(V_\delta)$ (the stack of mice over $X$, $V_\delta$ respectively)?

Suzuki’s Fact:\
${\mathsf{ZF}}\Rightarrow$ no parameter-definable $j:V\to V$
============================================================

The following fact is [@kanamori Proposition 5.1] (though it is stated under the assumption that $M,N$ are transitive proper class inner models there). We will need to prove analogues later, so let us look at this proof first:

\[fact:ZF\_Sigma\_1-elem\_cofinal\_implies\_full\_elem\] Let $M,N$ be models of ${\mathsf{ZF}}$. Let $j:M\to N$ be $\Sigma_1$-elementary and $\in$-cofinal. Then $j$ is fully elementary.

We prove by induction on $n<{\omega}$, that $j$ is $\Sigma_n$-elementary. Because $j$ is $\Sigma_1$-elementary, we have $j(V_\alpha)=V_{j(\alpha)}$ for each $\alpha$.

Suppose $j$ is $\Sigma_n$-elementary where $n\geq 1$. Let $C_n{\subseteq}{\mathrm{OR}}^M$ be the $M$-club class of all $\alpha$ such that $V^M_\alpha{\preccurlyeq}_n V^M_\lambda$ ($C_n$ is as defined over $M$, without parameters). Let $\alpha\in C_n$. Then $j(\alpha)\in C^N_n$ (defined analogously over $N$). For suppose $N{\models}\varphi(x)$ where $x\in V_{j(\alpha)}^N$ and $\varphi$ is $\Sigma_n$, but that $V_{j(\alpha)}^N{\models}\neg\varphi(x)$. The existence of such an $x$ is a $\Sigma_n$ assertion about the parameter $V_{j(\alpha)}^N$, satisfied by $N$, so $M$ satisfies the same about $V_\alpha^M$ (by $\Sigma_n$-elementarity of $j$). But $\alpha\in C_n$, contradiction.

Now suppose that $N{\models}\varphi(j(x))$, where $\varphi$ is $\Sigma_{n+1}$. Then by the $\in$-cofinality of $j$ and the previous remarks, we may pick $\alpha\in C_n$ such that $x\in V_\alpha^M$ and $V_{j(\alpha)}^N{\models}\varphi(j(x))$. But then $V_\alpha^M{\models}\varphi(x)$, and since $\alpha\in C_n$, it follows that $M{\models}\varphi(x)$, as desired.

Suzuki [@suzuki_no_def_j] established the following basic fact. We will also used variants of its proof later, and the proof is short, so we include it:

\[fact:suzuki\_no\_def\_j\] Assume ${\mathsf{ZF}}$. Then no class $k$ which is definable from parameters is an elementary $k:V\to V$.

(Of course, the theorem is really a theorem scheme, giving one statement for each possible formula $\varphi$ being used to define $k$ (from a parameter).)

Suppose that $k:V\to V$ is elementary and there is a $\Sigma_n$ formula $\varphi$ and $p\in V$ such that $k=\{(x,y)\bigm|\varphi(p,x,y)\}$. Now let $\kappa$ be the least critical point of a $\Sigma_1$-elementary $j:V\to V$ which is defined in this way from $\varphi$ and some parameter $q$; note that every such $j$ is also $\in$-cofinal. This defines the ordinal $\kappa$. Let $j:V\to V$ witness this, i.e. with ${\mathrm{cr}}(j)=\kappa$. Hence $\kappa\notin{\mathrm{rg}}(j)$. But by Lemma \[fact:ZF\_Sigma\_1-elem\_cofinal\_implies\_full\_elem\], in fact $j$ is fully elementary. Hence $\kappa\in{\mathrm{rg}}(j)$, contradiction.

Embeddings $j:V_\delta\to 
V_\delta$ {#sec:j:V_delta_to_V_delta}
==========================

In contrast to Suzuki’s Fact \[fact:suzuki\_no\_def\_j\] above, we have the following well-known fact:

\[fact:j\_def\_from\_frag\] Let $j:V_{\lambda+1}\to V_{\lambda+1}$ be $\Sigma_1$-elementary where $\lambda$ is a limit ordinal. Then $j$ is definable over $V_{\lambda+1}$ from the parameter $p=j{\!\upharpoonright\!}V_\lambda\in V_{\lambda+1}$.

Given $A\in V_{\lambda+1}$, note that $j(A)=\bigcup_{\alpha<\lambda}j(A{\cap}V_\alpha)$.

Given these two example cases, it is natural to consider analogous questions regarding elementary $j:V_\delta\to V_\delta$ for $\delta\in{\mathrm{OR}}$. Can such a $j$ be definable from parameters over $V_\delta$? (Clearly if $V_\delta{\models}{\mathsf{ZF}}$ then the answer is no, but we consider arbitrary ordinals $\delta$.) More generally, how close can such a $j$ be to $V_\delta$? For example, can we have $j\in L(V_\delta)$? Or $j\in{\mathrm{HOD}}(V_\delta)$? In such cases, can we also have that $(V_\delta,j){\models}{\mathrm{ZFR}}$? One can also vary the question by weakening the elementarity demand on $j$, or strengthening it, in that we demand that $j$ extends to a (somewhat) elementary $j^+:{\mathcal{J}}_\alpha(V_\delta)\to{\mathcal{J}}_\alpha(V_\delta)$, for example.

By the following theorem, if $\delta$ is a limit and $j:V_\delta\to V_\delta$ is elementary, then $j$ is not definable from parameters over $V_\delta$; actually we will prove a better version in Theorem \[tm:no\_def\_Sigma\_1\_V\_delta\_to\_V\_delta\], in which we only assume $\Sigma_1$-elementarity of $j$ (and still obtain a contradiction).

\[tm:no\_def\_V\_delta\_to\_V\_delta\] Let $j:V_\delta\to V_\delta$ be elementary and $p\in V_\delta$ and suppose that $j$ is definable over $V_\delta$ from the parameter $p$. Then $\delta=\beta+1$ is a successor and $\operatorname{rank}(p)=\beta$.

Suppose not. Fix $k<{\omega}$ and a $\Sigma_k$ formula $\varphi$ such that for some $\beta<\delta$ and $p\in V_\beta$, the set $$j_p=\{(x,y)\in V_\delta\bigm|V_\delta{\models}\varphi(p,x,y)\}$$ is such that $$j_p:V_\delta\to V_\delta\text{ and is fully elementary}.$$ Let $\mu_0$ be the least possible critical point of all such (fully elementary) embeddings $j_p$ (minimizing over all $p\in V_\beta$). Let $p_0\in V_\beta$ and $j_{p_0}$ witness this, so ${\mathrm{cr}}(j_{p_0})=\mu_0$.

For $n<{\omega}$, say that $q\in V_\beta$ is *$n$-good* iff $$j_q:V_\delta\to V_\delta\text{ and is }\Sigma_n\text{-elementary.}$$ Let $$A_n=\{q\in V_\beta\bigm|q\text{ is }n\text{-good}\}.$$ So $A_n\in V_\delta$ and note that $A_n$ is definable over $V_\delta$ from the parameter $\beta$.

Since $j=j_{p_0}$ is fully elementary, $j(A_n){\cap}V_\beta=A_n$. (We might have $j(\beta)>\beta$, but note this is okay.) Let $A=\bigcap_{n<{\omega}}A_n$, so also $A\in V_\delta$. Note that for every $q\in A$, $j_q:V_\delta\to V_\delta$ is fully elementary.

The sequence $\left<A_n\right>_{n<{\omega}}$ can easily be coded by a set in $V_\delta$ (or is literally in $V_\delta$ if $\delta$ is a limit), and therefore $$j(A)=\bigcap_{n<{\omega}}j(A_n),$$ so $p_0\in j(A)$. Therefore $V_\delta{\models}$“$\exists q\in j(A)$ such that ${\mathrm{cr}}(j_q)<j(\mu_0)$” (as witnessed by $p_0$). Pulling this back with the elementarity of $j$, $V_\delta{\models}$“$\exists q\in A$ such that ${\mathrm{cr}}(j_q)<\mu_0$”. But this contradicts the minimality of $\mu_0$.

Let $\xi$ be a limit ordinal, $n<{\omega}$ and $j:V_{\xi+n+2}\to V_{\xi+n+2}$ be elementary. Then $j$ is not definable over $V_{\xi+n+2}$ from $j{\!\upharpoonright\!}V_{\xi+1}$.

If $n>0$ then this is immediate by Theorem \[tm:no\_def\_V\_delta\_to\_V\_delta\]. Suppose $n=0$ and $j$ is definable over $V_{\xi+2}$ from $j{\!\upharpoonright\!}V_{\xi+1}$. Then by Fact \[fact:j\_def\_from\_frag\], in fact $j$ is definable over $V_{\xi+2}$ from $j{\!\upharpoonright\!}V_\xi$. This contradicts Theorem \[tm:no\_def\_V\_delta\_to\_V\_delta\].

Let $j:V_{\xi+n+3}\to V_{\xi+n+3}$ be elementary, where $n<{\omega}$. Can $j$ be definable over $V_{\xi+n+3}$ from $j{\!\upharpoonright\!}V_{\xi+n+2}$?

More generally, let $j:V_{\xi+n+2}\to V_{\xi+n+2}$ be elementary, where $n<{\omega}$. Can $j$ be definable over $V_{\xi+n+2}$ from some parameter (of rank $\xi+n+1$)?

We will now consider the case that $\delta$ is a limit and $j:V_\delta\to V_\delta$ is just $\Sigma_1$-elementary.

Let $\delta$ be a limit and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. For $A{\subseteq}V_\delta$, define $$j(A)=\bigcup_{\alpha<\delta}j(A{\cap}V_\alpha).$$ Define $j^1=j$ and for $n\geq 1$ define $j^{n+1}=j(j^n)$. For $x\in V_\delta$, we say that $x$ is *$(j,n)$-stable* iff $j^m(x)=x$ for all $m\in[n,{\omega})$.

Before we state the next theorem, we state what will be an immediate corollary:

Assume ${\mathsf{ZF}}$, let $\delta\in{\mathrm{OR}}$ be a limit and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. Suppose that either $V_\delta$ is inaccessible, or $\delta$ is singular and $j(\operatorname{cof}(\delta))=\operatorname{cof}(\delta)$. Then $j$ is fully elementary. In fact, for every $A{\subseteq}V_\delta$, $$j:(V_\delta,A)\to(V_\delta,j(A))$$ is fully elementary (with respect to ${\mathcal{L}}_{\dot{A}}$).

\[tm:an\_iterate\_is\_elementary\] Assume ${\mathsf{ZF}}$, let $\delta\in{\mathrm{OR}}$ be a limit and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. Then:

1.  \[item:j\_pres\_V\_alpha\] $j(V_\alpha)=V_{j(\alpha)}$ for all $\alpha<\delta$.

2.  \[item:j\^n\_pres\_Sigma\_1-elem\] Every $j^n:V_\delta\to V_\delta$ is $\Sigma_1$-elementary (and $\in$-cofinal); in fact, for each $A{\subseteq}V_\delta$, $$j^n:(V_\delta,A)\to(V_\delta,j^n(A))$$ is $\Sigma_1$-elementary in the langauge ${\mathcal{L}}_{\dot{A}}$ with symbols $\in,\dot{A}$, where $\dot{A}$ is a one-place predicate symbol which interprets the predicate.

3.  \[item:x\_stable\] If $x\in V_\delta$ and $j^n(x)=x$ then $x$ is $(j,n)$-stable.

4.  \[item:alpha\_gets\_stable\] For each $\alpha<\delta$ there is $n<{\omega}$ such that $\alpha$ is $(j,n)$-stable.

5.  \[item:P\_xi\_gets\_stable\] For each $\alpha<\delta$ and $\xi\in{\mathrm{OR}}$, letting $P$ be the set of all prewellorders of $V_\alpha$ of length $\xi$, there is $n<{\omega}$ such that $P$ is $(j,n)$-stable.

6.  \[item:j\^n\_fully\_elem\] Suppose that either:

    1.  $V_\delta$ is inaccessible, or

    2.  $V_\delta$ is non-inaccessible and $\delta$ is regular and ${\mathrm{codcof}}(\delta)$ is $(j,n)$-stable,[^3] or

    3.  $\delta$ is singular and $\operatorname{cof}(\delta)$ is $(j,n)$-stable.

    Then $j^n:V_\delta\to V_\delta$ is fully elementary, and in fact, for each, $A{\subseteq}V_\delta$, $$j^n:(V_\delta,A)\to(V_\delta,j(A))$$ is fully elementary.

Part \[item:j\_pres\_V\_alpha\]: We have $\left<V_\beta\right>_{\beta\leq\alpha}\in V_\delta$ and note that $j(\left<V_\beta\right>_{\beta\leq\alpha})=\left<V_\beta\right>_{\beta\leq 
j(\alpha)}$, by $\Pi_1$-elementarity.

Part \[item:j\^n\_pres\_Sigma\_1-elem\]: Let $\alpha<\delta$ and $\alpha'=j(\alpha)$ and $j'=j{\!\upharpoonright\!}V_\alpha$. So $j':V_\alpha\to V_{\alpha'}$ is fully elementary. This fact is preserved by $j$, by $\Sigma_1$-elementarity. Clearly also $j(j):V_\delta\to V_\delta$, and is therefore $\Sigma_0$-elementary with respect to these models. But $j(j)$ is also $\in$-cofinal, hence $\Sigma_1$-elementary (with respect to $\in$).

For the $\Sigma_1$-elementarity of $j^n:(V_\delta,A)\to(V_\delta,j^n(A))$, let $x\in V_\delta$ and $\varphi$ be $\Sigma_0$ (in the expanded language), and suppose $$(V_\delta,j^n(A)){\models}\exists y\ \varphi(j^n(x),y).$$ Let $\alpha<\delta$ be sufficiently large that $x\in V_\alpha$ and $$(V_{j^n(\alpha)},j^n(A){\cap}V_{j^n(\alpha)}){\models}\exists y\ 
\varphi(j^n(x),y).$$ Then by the $\Sigma_1$-elementarity of $j^n$ (just in the language with $\in$), $$(V_\alpha,A{\cap}V_\alpha){\models}\exists y\ \varphi(x,y),$$ so $(V_\delta,A){\models}\exists y\ \varphi(x,y)$ as desired.

Part \[item:x\_stable\]: If $x=j(x)$ then $$j(x)=j(j(x))=j(j)(j(x))=j(j)(x).$$

Part \[item:alpha\_gets\_stable\]: Suppose not and let $\alpha<\delta$ be least otherwise. Let $\alpha_0=\alpha$ and $\alpha_{n+1}=j(\alpha_n)$ for $n<{\omega}$. Let $\lambda=\sup_{n<{\omega}}\alpha_n$.

Suppose first that $\lambda<\delta$. Note then that $j(\lambda)=\lambda$ since $\left<\alpha_n\right>_{n<{\omega}}\in 
V_\delta$. So $j{\!\upharpoonright\!}V_\lambda:V_\lambda\to V_\lambda$ is elementary and $$j((V_\lambda,j{\!\upharpoonright\!}V_\lambda))=(V_\lambda,j(j){\!\upharpoonright\!}V_\lambda).$$ Also then $j^n(\lambda)=\lambda$ for all $n\in[1,{\omega})$ and $$j((V_\lambda,j^n{\!\upharpoonright\!}V_\lambda))=j^n((V_\lambda,j^n{\!\upharpoonright\!}V_\lambda))=(V_\lambda,j^{n+1}{\!\upharpoonright\!}V_\lambda).$$ Also, the structure $(V_\lambda,\left<j^k{\!\upharpoonright\!}V_\lambda\right>_{k\geq n})$ is definable, uniformly in $n$, over $(V_\lambda,j^n{\!\upharpoonright\!}V_\lambda)$. Therefore $\alpha$ is uniformly definable over this structure. Let $\varphi$ be the defining formula, so for $x\in V_\lambda$, we have $$(V_\lambda,j^n{\!\upharpoonright\!}V_\lambda){\models}\varphi(x)\iff x=\alpha.$$ This holds in particular for $n=1$. But then by the elementarity of $j$, we have $$(V_\lambda,j^2{\!\upharpoonright\!}V_\lambda){\models}\varphi(j(\alpha)),$$ but $j(\alpha)>\alpha$, a contradiction.

Now suppose instead that $\lambda=\delta$. By part \[item:j\^n\_pres\_Sigma\_1-elem\], $j^n:(V_\delta,j^n)\to 
(V_\delta,j^{n+1})$ is $\Sigma_1$-elementary. For $m\in[1,{\omega})$ let $$A_m=\{\beta<\alpha\bigm|j^i(\beta)>\beta\text{ for all }i\leq 
m\}.$$ Then $A_m$ is definable over $V_{\alpha_m+4}$ from the parameter $j{\!\upharpoonright\!}V_{\alpha_{m+3}}$, and uniformly so in $m$. Let $A=\bigcap_{m<{\omega}}A_m$. Then in fact, $A=\emptyset$, by the minimality of $\alpha$. But note that $\left<A_m\right>_{m<{\omega}}\in 
V_\delta$ and $$j(A)=j(\bigcap_{m<{\omega}}A_m)=\bigcap_{m<{\omega}}j(A_m),$$ and (by the local computability $A_m$), $$j(A_m)=\{\beta<j(\alpha)\bigm|j^{i+1}(\beta)>\beta\text{ for all }i\leq 
m\}.$$ Since $j^k(x)=x$ for all $k\geq 1$ whenever $j(x)=x$, in fact $$j(A_m)=\{\beta<j(\alpha)\bigm|j^{i}(\beta)>\beta\text{ for all }i\leq 
m+1\}.$$ So note that $\alpha\in\bigcap_{m<{\omega}}j(A_m)$. So $\bigcap_{m<{\omega}}j(A_m)\neq\emptyset$, although $\bigcap_{m<{\omega}}A_m=\emptyset$, contradicting the elementarity of $j$.

Part \[item:P\_xi\_gets\_stable\]: By the above, there is $n_0$ such that $\alpha$ is $(j,n_0)$-stable. Now argue as in the previous part from $n_0$ onward, and from the parameter $\alpha$, identify the collection $P$ of prewellorders of $V_\alpha$ of the form $P=P_\xi$ for some ordinal $\xi$, with $\xi$ least such that for no $n\in[n_0,{\omega})$ is $j_n(P)=P$. (Here $\xi\geq\delta$ is possible; we are just using the local definability of “prewellorder” within $V_\delta$, and of comparisons of their lengths.) Then note that we always have $j_n(P_\xi)=P_{\xi'}$ with some $\xi'>\xi$, since $j_n(P_\zeta)=P_\zeta$ for $\zeta<\xi$. One can argue for a contradiction much as before.

Part \[item:j\^n\_fully\_elem\]: By renaming, we may assume that either $\gamma=\delta$ or $j(\gamma)=\gamma$. Let $A{\subseteq}V_\delta$. We must show that $$j:(V_\delta,A)\to(V_\delta,j(A))$$ is fully elementary. The $\Sigma_1$-elementarity is by Part \[item:j\^n\_pres\_Sigma\_1-elem\].

We now verify $\Sigma_2$-elementarity. Let $x\in V_\delta$ and $\varphi$ be $\Pi_1$ and suppose that $$(V_\delta,j(A)){\models}\exists y\varphi(j(x),y),$$ and let $\beta<\delta$ be such that some $y\in V_{j(\beta)}$ witnesses this.

Suppose first that $\gamma<\delta$; so we are assuming $j(\gamma)=\gamma$. Let $f:\gamma\to\delta$ be cofinal and increasing. For $\xi<\gamma$ let $$B_\xi=\{z\in V_\beta\bigm| (V_{f(\xi)},A{\cap}V_{f(\xi)}){\models}\varphi(x,z)\}.$$ Then note that $$j(B_\xi)=\{z\in V_{j(\beta)}\bigm|(V_{j(f(\xi))},j(A){\cap}V_{j(f(\xi))}){\models}\varphi(j(x),z)\}.$$ Therefore $y\in j(B_\xi)$, so in fact $$y\in \left(\bigcap_{\xi<\gamma}j(B_\xi)\right)\neq\emptyset.$$ As $\gamma<\delta$, we have $\left<B_\xi\right>_{\xi<\gamma}\in V_\delta$. Also, $$\xi_0<\xi_1\implies B_{\xi_1}{\subseteq}B_{\xi_0}.$$ So the same holds of $j(\left<B_\xi\right>_{\xi<\gamma})$, and since $j(\gamma)=\gamma$, we have $j``\gamma$ cofinal in $j(\gamma)$, and so letting $j(\left<B_\xi\right>_{\xi<\gamma})=\left<B'_\xi\right>_{\xi<\gamma}$, $$j\left(\bigcap_{\xi<\gamma}B_\xi\right)=\bigcap_{\xi<\gamma}B'_\xi=
\bigcap_{\xi<\gamma} B'_{j(\xi)}=\bigcap_{\xi<\gamma}j(B_\xi)\neq\emptyset.$$ So $\bigcap_{\xi<\gamma}B_\xi\neq\emptyset$. But letting $z\in\bigcap_{\xi<\gamma}B_\xi$, note that $$(V_\delta,A){\models}\varphi(x,z),$$ as desired.

Now suppose instead that $\gamma=\delta$. Define $\left<B_\xi\right>_{\xi<\delta}$ as before, except that now $f(\xi)=\xi$ for $\xi<\delta$. If there is $\xi_0<\delta$ such that $B_\xi=B_{\xi_0}$ for all $\xi\in[\xi_0,\delta)$, then we easily have that $B_{\xi_0}\neq\emptyset$, and any $z\in B_{\xi_0}$ witnesses $\exists y\varphi(x,y)$ as before. Now suppose there is no such $\xi_0$. Given $z_0,z_1\in B=\bigcup_{\xi<\delta}B_\xi$, say that $z_0<^*z_1$ iff there is $\xi<\delta$ such that $z_1\in B_\xi$ but $z_0\notin B_\xi$. Then $<^*$ is a prewellorder on $B$, and $<^*$ is in $V_\delta$, and because $\gamma=\delta$ and there is no $\xi_0$ as above, $\delta$ is the the ordertype of $<^*$. So let $P$ be as in the statement of the lemma; then by assumption $j(P)=P$, which easily gives that $j({<^*})$ also has ordertype $\delta$. The function $z\mapsto B_{\operatorname{rank}(z)}$, with domain $B$, and where $\operatorname{rank}(z)$ is the ${<^*}$-rank of $z$, is also in $V_\delta$. But then we can argue as before to show that $\bigcap_{\xi<\delta}B_\xi\neq\emptyset$, which suffices, also as before.

So we have $\Sigma_2$-elementarity (with respect to an arbitrary $A{\subseteq}V_\delta$). Now suppose we have $\Sigma_k$-elementarity where $k\geq 2$. Let $$T=T^A_{k-1}=\operatorname{Th}_{\Sigma_{k-1}}^{(V_\delta,A)}(V_\delta);$$ that is, the set of all pairs $(\varphi,x)$ such that $\varphi$ is $\Sigma_{k-1}$ and $(V_\delta,A){\models}\varphi(x)$. By the $\Sigma_k$-elementarity of $j$, $$j(T)=\operatorname{Th}_{\Sigma_{k-1}}^{(V_\delta,j(A))}(V_\delta).$$ For given $\alpha<\delta$, we have $$(V_\delta,A){\models}{\forall}x\in 
V_\alpha\ [{\forall}\varphi\in({\mathcal{L}}_{\dot{A}}{\cap}\Sigma_{k-1})\ 
[\varphi(x)\iff (\varphi,x)\in T{\cap}V_\alpha]],$$ which is a $\Pi_k$ assertion of parameter $(V_\alpha,T{\cap}V_\alpha)$, which therefore lifts to $(V_\delta,j(A))$ regarding the parameter $(V_{j(\alpha)},j(T){\cap}V_{j(\alpha)})$.

So by what we have proved above, but with $T$ replacing $A$, we have that $j$ is $\Sigma_2$-elementary as a map $$\label{eqn:j_Sigma_2-elem_for_T} j:(V_\delta,(A,T))\to 
(V_\delta,(j(A),j(T))).$$

Now let $\varphi$ be $\Sigma_{k-1}$ and suppose that $$(V_\delta,j(A)){\models}\exists y{\forall}z\ [\varphi(j(x),y,z)];$$ equivalently, $$(V_\delta,(j(A),j(T))){\models}\exists y{\forall}z\ [(\varphi,(j(x),y,z))\in j(T)].$$ By the $\Sigma_2$-elementarity of $j$ with respect to the structures in line (\[eqn:j\_Sigma\_2-elem\_for\_T\]) above, therefore $$(V_\delta,(A,T)){\models}\exists y{\forall}z\ [(\varphi,(x,y,z))\in T];$$ equivalently, $(V_\delta,A){\models}\exists y{\forall}z\ [\varphi(x,y,z)]$, as desired.

Can there be a $\Sigma_1$-elementary $j:V_\delta\to V_\delta$, where $\delta$ is a limit, such that $j$ is not fully elementary? If ${\mathrm{ZFR}}$ is consistent then there can, and in fact with $j$ not $\Sigma_2$-elementary. For assume $(V,j){\models}{\mathrm{ZFR}}$. Let $\kappa={\mathrm{cr}}(j)$ and $\lambda=\lambda_j$. Then we get an example with $\delta=\lambda+\kappa$ and $k=j{\!\upharpoonright\!}V_\delta$. For since $j(\lambda)=\lambda$ and $j{\!\upharpoonright\!}\kappa={\mathrm{id}}$, we have $k:V_\delta\to V_\delta$, and clearly $k$ is $\in$-cofinal and $\Sigma_0$-elementary, hence $\Sigma_1$-elementary. But consider the $\Pi_2$ formula $$\varphi(\dot{\kappa},\dot{\lambda})=\text{``}
{\forall}\alpha<\dot{\kappa}\ 
\exists\xi\in{\mathrm{OR}}\ [\xi=\dot{\lambda}+\alpha]\text{''}.$$ Then $V_{\lambda+\kappa}{\models}\varphi(\kappa,\lambda)$, but $V_{\lambda+\kappa}{\models}\neg\varphi(j(\kappa),j(\lambda))$; that is, $V_{\lambda+\kappa}{\models}\neg\varphi(j(\kappa),\lambda)$, since $\alpha=\kappa<j(\kappa)$, but $\lambda+\kappa\not\in V_{\lambda+\kappa}$. For this example, $k^2(\kappa)=\kappa=\operatorname{cof}(\lambda+\kappa)$, so $k^2$ is fully elementary.

Can there be a $\Sigma_1$-elementary $j:V_\delta\to V_\delta$, with $\delta$ a limit, such that $j^2$ is not fully elementary?

Using the preceding theorem, we can now improve on Theorem \[tm:no\_def\_V\_delta\_to\_V\_delta\]:

\[tm:no\_def\_Sigma\_1\_V\_delta\_to\_V\_delta\] Assume ${\mathsf{ZF}}$ and let $\delta$ be a limit. Then there is no $j:V_\delta\to V_\delta$ which is $\Sigma_1$-elementary and definable from parameters over $V_\delta$.

Suppose otherwise. Then by Theorem \[tm:an\_iterate\_is\_elementary\], there is $n<{\omega}$ such that $j^n:V_\delta\to V_\delta$ is fully elementary, and since $j$ is definable from parameters over $V_\delta$, so is $j^n$. So fix formula $\varphi$ and $p\in V_\delta$ with $$j^n(x)=y\iff V_\delta{\models}\varphi(p,x,y)$$ for all $x,y\in V_\delta$.

Given $q\in V_\delta$, let $j_q=\{(x,y)\bigm| V_\delta{\models}\varphi(q,x,y)\}$. Now let $\kappa_0$ be the least critical point of any fully elementary $j_q:V_\delta\to V_\delta$. Let $q_0$ witness this. Let $\beta<\delta$ such that $(q_0,\kappa_0)\in V_{j_{q_0}(\beta)}$.

For $n<{\omega}$ and $(q,\mu)\in V_\beta$ with $\mu<\kappa_0$, say that $(q,\mu)$ is *$n$-good* iff $j_q:V_\delta\to V_\delta$ and is $\Sigma_{n+1}$-elementary with ${\mathrm{cr}}(j_q)=\mu$. Let $A_n$ be the set of all $n$-good pairs. So $\bigcap_{n<{\omega}}A_n=\emptyset$, by the minimality of $\kappa_0$. But we have $\left<A_n\right>_{n<{\omega}}\in V_\delta$, so $$\emptyset=j_{q_0}(\bigcap_{n<{\omega}}A_n)=\bigcap_{n<{\omega}}j_{q_0}(A_n).$$ But note that $(q_0,\kappa_0)\in j_{q_0}(A_n)$ for each $n<{\omega}$, because $\beta$ is large enough and $j_{q_0}(\kappa_0)>\kappa_0$ and $j_{q_0}$ is fully elementary. This is a contradiction.

We finish by observing that we can carry out most of the arguments above more locally, using just the theory ${\mathsf{RL}}$ below, which is the basic first order theory modelled by $V_\lambda$ for limit $\lambda>{\omega}$, assuming ${\mathsf{ZF}}$:

${\mathsf{RL}}$ (for *Rank Limit*) is the theory in ${\mathcal{L}}$ consisting of Extensionality, Foundation, Pairing, Power Set, Infinity, Separation (for all formulas, from parameters), together with the statements “For every ordinal $\alpha$, $\alpha+1$ exists, $V_\alpha$ exists and $\left<V_\beta\right>_{\beta<\alpha}$ exists”, and “For every $x$, there is an ordinal $\alpha$ such that $x\in V_\alpha$”.

Clearly a model $M{\models}{\mathsf{RL}}$ can contain objects $R$ such that $M{\models}$“$R$ is a wellorder”, but such that there is no $\alpha\in{\mathrm{OR}}^M$ such that $M{\models}$“$\alpha$ is the ordertype of $R$”.

We will now show that Suzuki’s Fact \[fact:suzuki\_no\_def\_j\] (the proof of which used ${\mathsf{ZF}}$) is actually a consequence of ${\mathsf{RL}}$:

Assume ${\mathsf{RL}}$. Then there is no $\Sigma_1$-elementary $j:V\to V$ which is definable from parameters.

By Suzuki’s fact, we may assume that ${\mathsf{ZF}}$ fails. So either (i) ${\mathrm{OR}}$ is ${\utilde{\Sigma}}_k$-singular for some $k$; that is, there is an ordinal $\gamma$ and $\Sigma_k$ formula $\varphi$ and parameter $p$ such that $\varphi(p,\xi,\beta)$ defines a cofinal map $\gamma\to{\mathrm{OR}}$, via $\xi\mapsto\beta$, and we take then $(\gamma,k)$ then lexicographically least, so $\gamma$ is regular; or (ii) otherwise, and there is $\eta\in{\mathrm{OR}}$ and $k<{\omega}$ and $\Sigma_k$ formula $\varphi$ and $p\in V$ such that $\varphi(p,x,\beta)$ defines a cofinal map $f:V_\eta\to{\mathrm{OR}}$, via $x\mapsto\beta$, and we take $\eta$ least such, which as before implies that $\eta=\alpha+1$ and that ${\mathrm{rg}}(f)$ has ordertype ${\mathrm{OR}}$, and then we may in fact take $f$ to be surjective, and obtain prewellorders of $V_{\alpha+1}$ having ordertype ${\mathrm{OR}}$, and so define $P={\mathrm{codcof}}({\mathrm{OR}})$ as before. (${\mathsf{RL}}$ suffices here; for example, starting with an arbtirary definable cofinal $f:V_{\alpha+1}\to{\mathrm{OR}}$, for each $\beta\in{\mathrm{OR}}$, ${\mathrm{rg}}(f){\cap}\beta$ is a set, by Separation. And the fact that a prewellorder has ordertype ${\mathrm{OR}}$ is a first-order statement under ${\mathsf{RL}}$.) Let $x_0$ be this $\gamma$ or $P$ respectively. Note that $x_0$ is definable without parameters (but maybe not $\Sigma_1$-definable without parameters).

Now as before, we can define each $j^n$, and $$j^n:(V,j^n)\to(V,j^{n+1})$$ is $\in$-cofinal $\Sigma_1$-elementary. Basically as before, one shows that for each ordinal $\alpha$, there is $n<{\omega}$ such that $\alpha$ is $(j,n)$-stable; however, we must use here the definability of $j$ from parameters and Separation to get that the relevant sequence of sets $\left<A_n\right>_{n<{\omega}}$ is a set; hence we can consider $A=\bigcap_{n<{\omega}}A_n$ and $j(A)$. Similarly, if ${\mathrm{OR}}$ is regular to class functions definable from parameters and $P$ is as above, then $P$ is $(j,n)$-stable for some $n$.

Now given any $\Sigma_1$-elementary $j:V\to V$, we have that $j$ is fully elementary iff $j(x_0)=x_0$. For if $j$ is elementary then $j(x_0)=x_0$ because $x_0$ is outright definable. The other direction is proved like before, but we must restrict to classes $A{\subseteq}V$ which are definable from parameters over $V$ (which clearly suffices for our purposes here), since we need to use Separation to get that the relevant $x_0$-indexed sequence of sets $\left<A_y\right>_y$ is a set.

Now assume there is some ${\utilde{\Sigma}}_k$-definable $j:V\to V$ which is $\Sigma_1$-elementary. Then by the above, there is such a $(k,j)$ satisfying $j(x_0)=x_0$. Let $\varphi$ be a $\Sigma_k$ formula for defining some such $j$. For $p\in V$ let $$j_p=\{(x,y)\bigm|\varphi(p,x,y)\}.$$ Let $\kappa_0\in{\mathrm{OR}}$ be least such that for some $p$, $j_p:V\to V$ is $\Sigma_1$-elementary and ${\mathrm{cr}}(j)=\kappa_0$ and $j_p(x_0)=x_0$. Since $x_0$ is outright definable, so is $\kappa_0$.

Fix $p_0$ witnessing this. Since $j_{p_0}$ is $\Sigma_1$-elementary and $j_{p_0}(x_0)=x_0$, and by the discussion above, $j_{p_0}$ is fully elementary. But $j_{p_0}(\kappa_0)>\kappa_0$, so $\kappa_0\notin{\mathrm{rg}}(j_{p_0})$, contradicting the definability of $\kappa_0$.

$j:V_\delta\to V_\delta$ in $L_\lambda(V_\delta)$ {#sec:j:V_delta_to_V_delta_in_L(V_delta)}
=================================================

In this section we consider the possibility of the existence of elementary (and $\Sigma_1$-elementary) $j:V_\delta\to V_\delta$ in the lower levels of the $L(V_\lambda)$ hierarchy, for limit $\delta$, and variants of this hypothesis.

In the following proof, we use Jensen’s refinement of the ${\mathcal{J}}$-hierarchy into the ${\mathcal{S}}$-hierarchy. Here is a summary of the features we need; the reader can refer to [@jensen_fs] or [@schindler2010fine] for more details. Recall that for a transitive set $X$, $${\mathcal{J}}(X)=\{f(V_\lambda,\vec{x})\bigm|f\text{ is a rudimentary function and 
}\vec{x}\in X^{<{\omega}}\}$$ $$=\bigcup_{n<{\omega}}{\mathcal{S}}_n(X),$$ where ${\mathcal{S}}$ is Jensen’s ${\mathcal{S}}$-operator, and ${\mathcal{S}}_0(X)=X$ and ${\mathcal{S}}_{n+1}(X)={\mathcal{S}}({\mathcal{S}}_n(X))$. The rudimentary functions are combinations of simple set operations, such as union, pairing, projection, etc. They are defined via a scheme, and each determined by one of countably many terms. Taking ${\mathcal{S}}$ defined appropriately (maybe not exactly how Jensen originally defined it), each ${\mathcal{S}}_n(X)$ is transitive, ${\mathcal{S}}_n(X)\in{\mathcal{S}}_{n+1}(X)$ and so ${\mathcal{S}}_n(X){\subseteq}{\mathcal{S}}_{n+1}(X)$. So in particular, $${\mathcal{S}}_n(X){\preccurlyeq}_0{\mathcal{S}}_{n+1}(X){\preccurlyeq}_0{\mathcal{J}}(X),$$ and for $\Sigma_1$ formulas $\varphi$ and parameters $y\in{\mathcal{J}}(X)$, we have $${\mathcal{J}}(X){\models}\varphi(y)\iff\exists n<{\omega}\ 
[y\in{\mathcal{S}}_n(X){\models}\varphi(y)]$$ $$\iff\exists n<{\omega}{\forall}m\in[n,{\omega})\ [y\in{\mathcal{S}}_m(X){\models}\varphi(y)].$$ Moreover, the truth of $\Sigma_1$ statements over ${\mathcal{J}}(X)$ reduces uniformly recursively to countable disjunctions of statements over $X$. That is, there is a recursive function $(\varphi,\vec{f},n)\mapsto\psi=\psi_{\varphi,\vec{f},n}$ such that for all $\Sigma_1$ formulas $\varphi$ and all tuples $\vec{f}$ of rudimentary function terms and all $n<{\omega}$, $\psi=\psi_{\varphi,\vec{f},n}$ is a formula in ${\mathcal{L}}$, and for all $\vec{x}
\in X^{<{\omega}}$, we have $${\mathcal{J}}(X){\models}\varphi(\vec{f}(V_\lambda,\vec{x}))\iff\exists n<{\omega}\  
X{\models}\psi_{\varphi,\vec{f},n}(\vec{x}).$$ Also, for each rudimentary $f$, the graph $$\{(\vec{x},y)\bigm|\vec{x}\in X^{<{\omega}}\text{ and }y=f(V_\lambda,\vec{x})\}$$ is $\Sigma_1$-definable over ${\mathcal{J}}(X)$ from the parameter $X$, uniformly in $X$.

Finally, we have that $${\mathcal{P}}(X){\cap}{\mathcal{J}}(X)=\{A{\subseteq}X\bigm|A\text{ is definable over }X\text{ from 
parameters}\}.$$

We can now proceed to the next lemma:

\[lem:j\_extends\_to\_j\^+\] Assume ${\mathsf{ZF}}$ and let $\lambda\in{\mathrm{OR}}$ be a limit and $j:V_\lambda\to V_\lambda$ be fully elementary. Then there is a unique $$j^+:{\mathcal{J}}(V_\lambda)\to{\mathcal{J}}(V_\lambda)$$ such that $j{\subseteq}j^+$ and $j^+$ is $\Sigma_1$-elementary.

Clearly we must set $j^+(\lambda)=\lambda$, and hence also $j^+(V_\lambda)=V_\lambda$. And because $${\mathcal{J}}(V_\lambda)=\bigcup_{n<{\omega}}{\mathcal{S}}_n(V_\lambda)$$ $$=\{f(V_\lambda,\vec{x})\bigm|\vec{x}\in V_\lambda^{<{\omega}}\},$$ and since for each rudimentary $f$, the graph $$\{(\vec{x},y)\bigm|\vec{x}\in V_\lambda^{<{\omega}}\text{ and 
}y=f(V_\lambda,\vec{x})\}$$ is $\Sigma_1$ over ${\mathcal{J}}(V_\lambda)$ in the parameter $V_\lambda$, we are forced to set $$j^+(f(V_\lambda,\vec{x})) =f(V_\lambda,j(\vec{x}));$$ this gives uniqueness.

But the definition of $j^+$ above gives a well-defined, $\Sigma_0$-elementary function $$j^+:{\mathcal{J}}(V_\lambda)\to{\mathcal{J}}(V_\lambda).$$ This is by [@jensen_fs]: all rudimentary functions are simple, and hence for each $\Sigma_0$ formulas $\varphi$ and all rud functions $f_0,\ldots,f_{k-1}$, there is a formula $\varphi'$ (of arbitrary complexity) such that for all $\vec{x}\in V_\lambda^{<{\omega}}$, we have $${\mathcal{J}}(V_\lambda){\models}\varphi(f_0(V_\lambda,\vec{x}),
f_1(V_\lambda,\vec{x}),\ldots,
f_{k-1}(V_\lambda,\vec{x}))$$ iff $$V_\lambda{\models}\varphi'(\vec{x}).$$ But then by the elementarity of $j$, this is equivalent to $$V_\lambda{\models}\varphi'(j(\vec{x})),$$ in turn equivalent to $${\mathcal{J}}(V_\lambda){\models}\varphi(f_0(V_\lambda,j(\vec{x})),
 f_1(V_\lambda,j(\vec{x})),\ldots,f_{k-1}(V_\lambda,j(\vec{x}))).$$ The map $j^+$ is also $\in$-cofinal, and hence $\Sigma_1$-elementary. And note that $j{\subseteq}j^+$.

Note that in the following, we work with (true) $V_\lambda$, so in particular, $j{\!\upharpoonright\!}V_\alpha\in V_\lambda$ for each $\alpha<\lambda$. We do not assume that $V_\lambda{\models}{\mathsf{ZF}}$ (let alone $(V_\lambda,j){\models}{\mathsf{ZF}}_j$).

\[tm:j\_not\_amenably\_Sigma\_1\] Assume ${\mathsf{ZF}}$. Let $\lambda\in{\mathrm{OR}}$ be a limit. Suppose that $j:V_\lambda\to V_\lambda$ is elementary. Then $$\widetilde{j}=\{j{\!\upharpoonright\!}V_\alpha\bigm|\alpha<\lambda\}$$ is not ${\utilde{\Sigma}}_1^{{\mathcal{J}}(V_\lambda)}$.

Suppose otherwise. Recall that ${\mathcal{J}}(V_\lambda)=\bigcup_{n<{\omega}}{\mathcal{S}}_n(V_\lambda)$. Let $\kappa$ be the least critical point of such an embedding $j$, choose a $j$ witnessing this, and choose $p_0\in V_\lambda$ such that $\widetilde{j}$ is $\Sigma_1^{{\mathcal{J}}(V_\lambda)}(\{p,V_\lambda\})$. We may fix a $\Sigma_0$ formula $\varphi$ and a sequence $\left<\eta_n\right>_{n<{\omega}}{\subseteq}\lambda$, with $\eta_{n}\leq\eta_{n+1}$ and $\lambda=\bigcup_{n<{\omega}}\eta_n$, such that for each $n<{\omega}$, we have $${\mathcal{S}}_n(V_\lambda){\models}\text{``}j{\!\upharpoonright\!}V_{\eta_n}\text{ is the unique 
}k\text{ such that }\varphi(n,p,V_\lambda,k)\text{''}$$ (in particular, it follows that $\operatorname{cof}(\lambda)={\omega}$, definably from parameters over ${\mathcal{J}}(V_\lambda)$).

Let $(q,\mu)\in V_\lambda{\times}\lambda$ and $m<{\omega}$.

Say that $(q,\mu)$ is *$m$-good* iff for each $n\leq m$, there are $(\xi_n,\xi'_n)\in\lambda^2$ and $\ell_n\in V_\lambda$ such that:

1.  $\ell_n:V_{\xi_n}\to V_{\xi_n'}$ is elementary,

2.  $\ell_n$ is $\Sigma_m$-elementary with respect to $V_\lambda\to V_\lambda$ (that is, for all $\Sigma_m$ formulas $\psi$, and all $x\in V_{\xi_n}$, we have $V_\lambda{\models}\psi(x)$ iff $V_\lambda{\models}\psi(\ell_n(x))$),

3.  ${\mathrm{cr}}(\ell_n)=\mu$,

4.  ${\mathcal{S}}_n(V_\lambda){\models}\text{``}\ell_n\text{ is the unique 
    }k\text{ such that }\varphi(n,q,V_\lambda,k)\text{''}$,

5.  $\ell_i{\subseteq}\ell_n$ for each $i\leq n$.

Write $\ell^{q}_n=\ell_n$ and $\xi^q_n=\xi_n$.

Let also $n<{\omega}$. Say that $(q,\mu)$ is *$(m,n)$-strong* iff $(q,\mu)$ is $m$-good and $\eta_n\leq\xi^q_m$. Say that $(q,\mu)$ is *$n$-strong* iff $\exists m<{\omega}\ [(q,\mu)\text{ is }(m,n)\text{-strong}]$.

Recall $j:V_\lambda\to V_\lambda$ is elementary with ${\mathrm{cr}}(j)=\kappa_0$, etc. Let $\alpha_0<\lambda$ be such that $p_0\in j(V_{\alpha_0})$. Let $$A_m=\{(q,\mu)\in V_{\alpha_0}{\times}\kappa_0\bigm|(q,\mu)\text{ is 
}m\text{-good}\}.$$

Let $A_{\omega}=\bigcap_{m<{\omega}}A_m$. Note that $\left<A_m\right>_{m<{\omega}}\in V_\lambda$.

By Lemma \[lem:j\_extends\_to\_j\^+\], because $j:V_\lambda\to V_\lambda$ is elementary, it extends uniquely to a $\Sigma_1$-elementary $$j^+:{\mathcal{J}}(V_\lambda)\to{\mathcal{J}}(V_\lambda)$$ such that $j^+(V_\lambda)=V_\lambda$. This gives that $j^+({\mathcal{S}}_n(V_\lambda))={\mathcal{S}}_n(V_\lambda)$ for each $n<{\omega}$, and $$j^+_n=j^+{\!\upharpoonright\!}{\mathcal{S}}_n(V_\lambda):{\mathcal{S}}_n(V_\lambda)\to{\mathcal{S}}_n(V_\lambda)$$ is fully elementary.

$A_{\omega}\neq\emptyset$, and moreover, $(p_0,\kappa_0)\in j(A_{\omega})$.

Because $j^+_n$ is fully elementary, and $A_n\in V_\lambda$, note that $$j(A_n)=j^+_n(A_n)=\{(q,\mu)\in V_{j(\alpha_0)}{\times}j(\kappa_0)\bigm|(q,\mu)\text{ is }n\text{-good}\}.$$ But $(p_0,\kappa_0)$ is $n$-good, so $$(p_0,\kappa_0)\in j(A_n).$$ Also $\left<A_n\right>_{n<{\omega}}\in V_\lambda$, and $$j(A_{\omega})=j(\bigcap_{n<{\omega}}A_n)=\bigcap_{n<{\omega}}j(A_n),$$ so $(p_0,\kappa_0)\in j(A_{\omega})\neq\emptyset$, so $A_{\omega}\neq\emptyset$.

Let $$B_{(m,n)}=\{(q,\mu)\in V_{\alpha_0}{\times}\kappa_0\bigm|(q,\mu)\text{ is 
}(m,n)\text{-strong}\},$$ and $$B_n=\{(q,\mu)\in V_{\alpha_0}{\times}\kappa_0\bigm|(q,\mu)\text{ is 
}n\text{-strong},\}$$ and $B_{\omega}=\bigcap_{n<{\omega}}B_n$. These sets are in $V_\lambda$.

$A_{\omega}{\cap}B_{\omega}\neq\emptyset$, and moreover, $(p_0,\kappa_0)\in 
j(A_{\omega}{\cap}B_{\omega})$.

By the previous claim and like in its proof, it suffices to see that $(p_0,\kappa_0)\in j(B_n)$ for each $n<{\omega}$. But $B_n=\bigcup_{m<{\omega}}B_{(m,n)}$, so $$j(B_n)=\bigcup_{m<{\omega}}j(B_{(m,n)}),$$ and $j(B_{(m,n)})=j^+_m(B_{(m,n)})=$ $$=\{(q,\mu)\in 
V_{j(\alpha_0)}{\times}j(\kappa_0)\bigm|(q,\mu)\text{ is 
}m\text{-good, with }j(\eta_n)\leq\xi^q_m\},$$ but there is $m<{\omega}$ such that $j(\eta_n)\leq\eta_m=\xi^{p_0}_m$, so with this $m$, we have $$(p_0,\kappa_0)\in j(B_{(m,n)}),$$ so $(p_0,\kappa_0)\in j(B_n)$, as required.

Now by the claims, we may pick $(q,\mu)\in A_{\omega}{\cap}B_{\omega}$. Let $$\ell=\bigcup_{n<{\omega}}\ell^q_n.$$ The final claim below contradicts the minimality of $\kappa_0$, completing the proof:

$\ell:V_\lambda\to V_\lambda$ is fully elementary, ${\mathrm{cr}}(\ell)=\mu<\kappa_0$, and $\widetilde{\ell}$ is ${\utilde{\Sigma}}_1^{{\mathcal{J}}(V_\lambda)}$.

Because $(q,\mu)\in A_{\omega}$, for each $n<{\omega}$, $$\ell^q_n:V_{\xi^q_n}\to V_{\xi_n'}$$ is fully elementary with ${\mathrm{cr}}(\ell^q_n)=\mu$, and $\ell^q_n{\subseteq}\ell^q_{n+1}$. So $\ell$ is a function with domain $$\left(\bigcup_{n<{\omega}}V_{\xi^q_n}\right)=V_\lambda;$$ the equality is because $(q,\mu)\in B_{\omega}$. Also each $\xi_n'<\lambda$, so $\bigcup_{n<{\omega}}\xi'_n=\lambda$, so $$\ell:V_\lambda\to V_\lambda.$$ Because $(q,\mu)\in A_{\omega}$, $\ell$ is easily $\Sigma_m$-elementary for each $m$, and hence is fully elementary. We also have ${\mathrm{cr}}(\ell)=\mu={\mathrm{cr}}(\ell^q_m)$ for each $m$.

Finally note that $\widetilde{\ell}$ is appropriately definable from the parameter $(q_0,V_\lambda)$.

\*\*\*The remainder of this section is just a sketch. What is written might assume that $V_\delta{\models}{\mathsf{ZF}}$ (writing $\delta$ now instead of what was $\lambda$ above), but that should no longer be important. I think that the arguments can be modified to show that one cannot have $j\in{\mathcal{J}}_\lambda(V_\delta)$, where $\lambda=\lambda_j$. Up to $\kappa$ there is not much difference, because the embedding extends to $j^+:{\mathcal{J}}_\kappa(V_\delta)\to{\mathcal{J}}_\kappa(V_\delta)$, fixing the ordinals $>\lambda$. Above here one wants to say that if some $j\in{\mathcal{J}}_\alpha(V_\delta)$ with $\alpha<\lambda_j$, then there is an iterate $j^n$ of $j$ which also gets in, and with ${\mathrm{cr}}(j^n)>\alpha$. From there, one has to just be a little careful with the “least critical point” business, because that could drop the critical point back $<\alpha$. Instead, look at the $j$ with the least possible critical point $>\alpha$, and then show that it’s actually a limit of such critical points $>\alpha$, by arguing much as before, for a contradiction.

Suppose there is a ${\utilde{\Sigma}}_1^{{\mathcal{J}}(V_\delta)}$ function $f:V_\alpha\to\delta$ for some $\alpha<\delta$, which is cofinal. Then there is in fact such a function with $f:{\omega}\to\delta$.

We have $f=\bigcup_{n<{\omega}}f_n$ where $f_n$ is the interpretation of the defining $\Sigma_1$ formula (in parameter) over ${\mathcal{S}}_n(V_\delta)$, and $f_n$ is definable from parameters over $V_\delta{\models}{\mathsf{ZF}}$. Therefore $f_n$ is bounded in $\delta$. But clearly the function $n\mapsto\sup(f_n)$ is also ${\utilde{\Sigma}}_1^{{\mathcal{J}}(V_\delta)}$, and this function is cofinal in $\delta$, as desired.

We now generalize to ${\utilde{{\mathrm{r}\Sigma}}}_2^{{\mathcal{J}}(V_\delta)}$. The ${\mathrm{r}\Sigma}$ complexity hierarchy is as in Mitchell-Steel fine structure. We have ${\mathcal{J}}(V_\delta)={\mathrm{Hull}}_1^{{\mathcal{J}}(V_\delta)}(V_\delta\cup\{V_\delta\})$. Let $T_1$ be the set of pairs $(t,q)\in{\mathcal{J}}(V_\delta)$ such that for some $\alpha<\delta$, we have $$t=\operatorname{Th}_1^{{\mathcal{J}}(V_\delta)}(V_\alpha\cup\{q\}).$$ Then an ${\mathrm{r}\Sigma}_2^{{\mathcal{J}}(V_\delta)}$ relation is one which is definable in the form $$\varphi(x)\iff\exists q,t\ [(t,q)\in T_1\text{ and }\psi(x,q,t)],$$ where $\psi$ is $\Sigma_1$.

\[rem:no\_j\_in\_J\_lambda(V\_delta)\] Suppose there is $j:V_\delta\to V_\delta$ which is elementary and amenably ${\utilde{{\mathrm{r}\Sigma}}}_2^{{\mathcal{J}}(V_\delta)}$. We have $j$ extends uniquely to $\Sigma_1$-elementary $j^+:{\mathcal{J}}(V_\delta)\to{\mathcal{J}}(V_\delta)$. Also $\rho_1^{{\mathcal{J}}(V_\delta)}=\delta$ and $j^+(t)=t'$ when $t=\operatorname{Th}_1^{{\mathcal{J}}(V_\delta)}(V_\xi\cup\{x\})$ and $t'=\operatorname{Th}_1^{{\mathcal{J}}(V_\delta)}(V_{\xi'}\cup\{x'\})$ where $(\xi',x')=j^+(\xi,x)$. And $j^+$ is $\in$-cofinal in $V_\delta$. It follows that $j^+$ is in fact ${{\utilde{{\mathrm{r}\Sigma}}}_2}$-elementary.

Now we may assume that the ${{\utilde{{\mathrm{r}\Sigma}}}_2}^{{\mathcal{J}}(V_\delta)}$-cofinality of $\delta$ is ${\omega}$; if ${{\utilde{{\mathrm{r}\Sigma}}}_1}^{{\mathcal{J}}(V_\delta)}$-cofinality of $\delta$ is ${\omega}$, this is trivial, and otherwise, by the lemma, $\delta$ is ${{\utilde{{\mathrm{r}\Sigma}}}_1}^{{\mathcal{J}}(V_\delta)}$-inaccessible. Therefore for each $\alpha<\delta$ and $x\in{\mathcal{J}}(V_\delta)$, ${\mathrm{Hull}}_1^{{\mathcal{J}}(V_\delta)}(V_\alpha\cup\{x,V_\delta\})$ is bounded in $\delta$, and if we let $\beta$ be its sup below $\delta$, note that ${\mathrm{Hull}}_1^{{\mathcal{J}}(V_\delta)}(V_\beta\cup\{x,V_\delta\})$ also has sup $\beta$ below $\delta$ (by breaking $\Sigma_1$ into the usual ${\omega}$-union), and if $\varphi$ is $\Sigma_1$ and $y\in V_\beta$ and ${\mathcal{J}}(V_\delta){\models}\exists z\varphi(x,y,V_\delta,z)$, then there is $z\in{\mathrm{Hull}}_1^{{\mathcal{J}}(V_\delta)}(V_\beta\cup\{x,V_\delta\})$ witnessing this, considering how things in ${\mathcal{J}}(V_\delta)$ are $\Sigma_1$-definable from ordinals and the parameter $V_\delta$ and parameters in $V_\delta$. Therefore this hull is $\Sigma_1$-elementary in ${\mathcal{J}}(V_\delta)$, and transitive $<\delta$.

Now fix some parameter $p\in{\mathcal{J}}(V_\delta)$ and ${\mathrm{r}\Sigma}_2$-formula $\varphi$ such that there is $j:V_\delta\to V_\delta$ elementary and $$\{j{\!\upharpoonright\!}V_\alpha\bigm|\alpha<\delta\}=\{k\bigm|{\mathcal{J}}(V_\delta){\models}\varphi(k,p)\}.$$ Define $\left<\beta_n,\alpha_n\right>_{n<{\omega}}$ by setting $\beta_0=0$, and then given $\beta_n$, set $\alpha_n<\delta$ to be the least $\alpha<\delta$ such that $\alpha\geq\beta_n$ and for some $$k:V_{\beta_n}\to 
V_{\beta'}$$ in $V_\delta$, $\operatorname{Th}_1^{{\mathcal{J}}(V_\delta)}(V_\alpha\cup\{k,V_\delta,p\})$ codes a witness to $\varphi(k,p)$, and then let $(\alpha_n,\beta_{n+1})$ be like the pair $(\alpha,\beta)$ in the discussion of the hull earlier, with $x=(p,\alpha_n)$.

Let $\beta=\sup_{n<{\omega}}\beta_n=\sup_{n<{\omega}}\alpha_n$, and $H={\mathrm{Hull}}_1^{{\mathcal{J}}(V_\delta)}(V_\beta\cup\{p,V_\delta\})$. Then we have $H{\cap}\delta=\beta$, $H{\preccurlyeq}_1{\mathcal{J}}(V_\delta)$, and letting $R$ be the transitive collapse, $\pi:R\to H$ the uncollapse, then $\pi(\beta)=\delta$, $V_\beta{\subseteq}R$, and $\pi$ is $\Sigma_1$-elementary, so $R={\mathcal{J}}(V_\beta)$. Thus, $\left<\alpha_n,\beta_n\right>_{n<{\omega}}$ is defined in the same way over $R$, giving that ${{\utilde{{\mathrm{r}\Sigma}}}_2}^{{\mathcal{J}}(V_\beta)}$-cofinality of $\beta$ is ${\omega}$, and note here that $j{\!\upharpoonright\!}V_\beta:V_\beta\to V_\beta$ is $\in$-cofinal and $\Sigma_1$-elementary, and is defined from $\varphi(\bar{p},\cdot)$ (where $\pi(\bar{p})=p$) over ${\mathcal{J}}(V_\beta)$. We also automatically have $V_\beta{\models}{\mathsf{ZF}}$, and in fact $\beta$ is ${{\utilde{\Sigma}}_1}^{{\mathcal{J}}(V_\beta)}$-regular, because we closed under enough hulls.

So we may assume that ${{\utilde{{\mathrm{r}\Sigma}}}_2}^{{\mathcal{J}}(V_\delta)}$-cofinality of $\delta$ is ${\omega}$. So fix a sequence $\left<\delta_n\right>_{n<{\omega}}$ witnessing this. Fix $\alpha<\delta$ such that $p_0\in V_\alpha$ and $\varphi$ an ${\mathrm{r}\Sigma}_2$ formula such that $\varphi(p_0,\cdot)$ defines an elementary $j:V_\delta\to V_\delta$; as discussed earlier, $j^+$ is ${\mathrm{r}\Sigma}_2$-elementary.

Given $q\in V_\alpha$ and $n<{\omega}$, say that $q$ is *$n$-good* iff $\operatorname{Th}_1^{{\mathcal{J}}(V_\delta)}(V_{\delta_n}\cup\{V_\delta\})$ does not code a witness demonstrating that $j_q$ fails, i.e. if we let $k$ be the union of the things it accepts, then $k:V_\beta\to V_{\beta'}$ for some $\beta\leq\beta'<\delta$, and $k$ is $\Sigma_0$-elementary. And say that $q$ is *$(n,m)$-strong* iff $\operatorname{Th}_1^{{\mathcal{J}}(V_\delta)}(V_{\delta_n}\cup\{V_\delta\})$ codes a witness demonstrating that $V_{\delta_m}{\subseteq}{\mathrm{dom}}(j_q)$. Both *$n$-good* and *$(n,m)$-strong* are each ${\mathrm{r}\Sigma}_2^{{\mathcal{J}}(V_\delta)}(\{a\})$-assertions, where the sequence $\left<\delta_n\right>_{n<{\omega}}$ is ${\mathrm{r}\Sigma}_2^{{\mathcal{J}}(V_\delta)}(\{a\})$. And letting $A_n$ be the set of all $q\in V_\alpha$ such that $q$ is $n$-good, then since $j^+$ preserves $\Sigma_1$-theories, we get that $j(A_n)$ is the set of all $q'\in V_{j(\alpha)}$ such that $\operatorname{Th}_1^{{\mathcal{J}}(V_\delta)}(V_{j(\delta_n)}\cup\{V_\delta\})$ does not code a bad witness for $q'$. So $j(A_n){\cap}V_\alpha{\subseteq}A_n$ (with equality if $j(\delta_n)=\delta_n$), and letting $A=\bigcap_{n<{\omega}}A_n$, clearly $j(A){\cap}V_\alpha= A$ is the set of $q\in V_\alpha$ such that $j_q$ is some $\Sigma_0$-elementary function $j_q:V_{\beta}\to V_{\beta'}$ for some $\beta\leq\beta'\leq\delta$. (since $\sup_{n<{\omega}}\delta_n=\sup_{n<{\omega}}j(\delta_n)=\delta$.) It works similarly for $B=\bigcap_{m<{\omega}}\bigcup_{n<{\omega}}B_{m,n}$, where $B_{m,n}$ is the set of $(m,n)$-strong $q\in V_\alpha$. Now finish the argument like before for a contradiction.

The same kind of argument should give that no $j:V_\delta\to V_\delta$ can be in ${\mathcal{J}}({\mathcal{J}}(V_\delta))$, and that should generalize a fair amount; the discussion earlier should give a proof that there is no such $j\in{\mathcal{J}}_\lambda(V_\delta)$. Beyond here, one seems to really want to start taking ultrapowers to extend the embeddings, which is what we do in the next section, but we need to assume that $\operatorname{cof}(\delta)>{\omega}$ to get that to work.

Ultrapowers and extenders under ${\mathsf{ZF}}$ {#sec:ultrapowers_in_ZF}
===============================================

In order to help us analyse the model $L(V_\delta)$ further, we want to be able to deal with ultrapowers via extenders, in order to assist us in extending embeddings of the form $j:V_\delta\to 
V_\delta$ to larger models, such as $L(V_\delta)$.

So we need to deal with extenders, and without ${\mathsf{AC}}$, there are some small technical difficulties here in the definitions, which we will work through first. The more significant issues are of course those of wellfoundedness of the ultrapower and (generalized) Los’ theorem; when we apply extenders, we will need to verify that these properties hold of the ultrapowers we form.

When we say that a set $a$ is *extensional*, we mean that $(a,\in{\!\upharpoonright\!}a)$ is extensional. Given a set $X$, we write $\langle X\rangle^{<{\omega}}$ for the set of all finite extensional subsets of $X$. Fix a recursive enumeration $\left<\bar{a}_n\right>_{n<{\omega}}$ of $V_{\omega}$. Let $a$ be a finite extensional set. The *signature* ${\mathrm{sig}}(a)$ of $a$ is $\bar{a}_n$ where $n$ is least such that $\bar{a}_n$ is extensional and $(\bar{a}_n,\in{\!\upharpoonright\!}\bar{a}_n)$ is isomorphic to $(a,\in{\!\upharpoonright\!}a)$. Given a signature $\bar{a}\in V_{\omega}$ (that is, $\bar{a}$ is extensional and ${\mathrm{sig}}(\bar{a})=\bar{a}$), and a transitive set $X$, we write $$\langle X\rangle^{\bar{a}}=\{b\in\langle 
X\rangle^{<{\omega}}\bigm|{\mathrm{sig}}(b)=\bar{a}\}.$$

Under ${\mathsf{ZF}}$, an easy induction on rank shows that for every finite set $a$ there is a finite set $b$ with $a{\subseteq}b$, of the same rank as $a$, and $b$ extensional. Note also that if $a$ is extensional then $a{\cap}V_\xi$ is also extensional.

Let $j:M\to N$ be elementary where $M,N$ model (enough) ${\mathsf{ZF}}$ and $\alpha\in{\mathrm{OR}}^M$ and $\beta=\sup j``\alpha$. Then the *$V_\beta^N$-extender $E_{j,\alpha,\beta}$ derived from $j$* is the set $E$ of all pairs $(A,a)$ such that:

1.  $a\in\langle V_\beta^N\rangle^{<{\omega}}$,

2.  $A\in M$ and $A{\subseteq}\langle V_\xi^M\rangle^{{\mathrm{sig}}(a)}$ for some $\xi<\alpha$,

3.  $a\in j(A)$.

Given a function $f$ with domain $\langle 
V_\alpha^M\rangle^{<{\omega}}$, say that $f$ has *support $\xi$* iff there is a function $f'$ with domain $\langle V_\xi^M\rangle^{<{\omega}}$ such that $$f(u)=f'(u{\cap}V_\xi)$$ for each $u$ (recall that if $u$ is extensional then so is $u{\cap}V_\xi$). We say that a function $f$ is *relevant* (with respect to $E$) iff $f$ has domain $\langle V_\alpha^M\rangle^{<{\omega}}$ and has support $\xi$ for some $\xi<\alpha$.

Let $f$ be a relevant function. Let $a,a'\in\langle 
V_\beta^M\rangle^{<{\omega}}$ with $a{\subseteq}a'$. Let $\xi$ be least such that $a'{\subseteq}V_{j(\xi)}^N$. We define a relevant function $f^{a,a'}$, with support $\xi$, as follows. Let $u''\in\langle V_\alpha^M\rangle^{<{\omega}}$ and let $u'=u''{\cap}V_{\xi}$. If ${\mathrm{sig}}(u')\neq{\mathrm{sig}}(a')$ then $f^{a,a'}(u')=\emptyset$. Suppose now that ${\mathrm{sig}}(u')={\mathrm{sig}}(a')$. Let $\pi:u'\to a'$ be the unique $\in$-isomorphism. Let $u=\pi^{-1}(a)$. Then $$f^{a,a'}(u'')=f^{a,a'}(u')=f(u).$$

Let $P{\models}{\mathsf{ZF}}$ with $\alpha\leq{\mathrm{OR}}^P$ and $V_\alpha^P=V_\alpha^M$. A *$P$-relevant pair* (with respect to $E$) is a pair $(a,f)$ such that $f$ is a relevant function, with $$f{\!\upharpoonright\!}\langle V_\xi^P\rangle^{<{\omega}}\in P,$$ where $\xi$ is a support of $f$ (hence if $\alpha<{\mathrm{OR}}^P$ then $f\in P$), and with $a\in\langle V_\beta^N\rangle^{<{\omega}}$.

We next define the *ultrapower ${\mathrm{Ult}}(P,E)$ of $P$ by $E$*. Given $P$-relevant pairs $(a,f),(b,g)$, say that $$(a,f)\approx_E(b,g)$$ iff letting $\xi<\alpha$ be some/any such that $a\cup b{\subseteq}V_{j(\xi)}^N$, and letting $c\in\langle V_{j(\xi)}^N\rangle^{<{\omega}}$ be some/any with $a\cup 
b{\subseteq}c$, then $$\{u\in\langle V_\xi^M\rangle^{{\mathrm{sig}}(c)}\bigm|f^{a,c}(u)=g^{b,c}(u)\}\in E_c.$$ We define the relation $$(a,f)\in_E(b,g)$$ likewise, replacing the condition “$f^{a,c}(u)=g^{b,c}(u)$” with “$f^{a,c}(u)\in f^{b,c}(u)$”.

We have:

1.  \[item:indep\_of\_xi,c\] In the definition of $\approx_E$ and $\in_E$, the choice of $\xi,c$ is irrelevant.

2.  $\approx_E$ is an equivalence relation on the $P$-relevant pairs,

3.  $\in_E$ respects $\approx_E$.

Part \[item:indep\_of\_xi,c\]: Consider $\approx_E$, and pairs $(a,f),(b,g)$. Take choices $\xi,c$ and $\xi',c'$ as in the definition. Note that it suffices to consider the case that $\xi\leq\xi'$ and $c{\subseteq}c'$. We must see that $$\{u\in\langle V_\xi^M\rangle^{{\mathrm{sig}}(c)}\bigm|f^{a,c}(u)=g^{b,c}(u)\}\in E_c$$ iff $$\{u\in\langle 
V_{\xi'}^M\rangle^{{\mathrm{sig}}(c')}\bigm|f^{a,c'}(u)=g^{b,c'}(u)\}\in E_{c'}.$$ But using that $c{\subseteq}c'$, this follows easily from the definitions.

The rest is straightforward.

We also define the associated *ultrapower embedding* $$i^P_E:P\to{\mathrm{Ult}}(P,E)$$ by setting $i^P_E(x)=(a,f)$ where $a=\emptyset$ and $f(u)=x$ for all $u$.

Of course, if $P\not{\models}{\mathsf{AC}}$ then the proof of Los’ theorem does not go through in the usual manner, so in general it is not clear that ${\mathrm{Ult}}(P,E)$ is even extensional. If ${\mathrm{Ult}}(P,E)$ is both extensional and wellfounded, then we identify it as usual with its transitive collapse.

Given $x\in V_\beta^N$, let $a_x=\{x\}$ and $f_x={\mathrm{id}}_\xi$, where $\xi$ be least such that $x\in V_{j(\xi)}^N$ and ${\mathrm{id}}_\xi$ the function with domain $[V_\xi^M]^{1}$ such that ${\mathrm{id}}_\xi(u)=u$ for each $u$. We have:

\[tm:extender\_coheres\] $V_\beta^N{\subseteq}$ the wellfounded part of ${\mathrm{Ult}}(P,E)$, and $V_\beta^N=V_\beta^{{\mathrm{Ult}}(P,E)}$. Moreover, for $x\in V_\beta^N$, we have $x=(a_x,f_x)$.

Easily from the definitions, we get that $$(a_x,f_x)\approx_E(a_y,f_y)\iff x=y,$$ $$(a_x,f_x)\in_E(a_y,f_y)\iff x\in y.$$ So let $(a,f)$ be a $P$-relevant pair such that $$(a,f)\in_E(a_x,f_x)$$ where $x\in V_\beta^N$. We may assume that $x\in a$. For simplicity assume $\alpha<{\mathrm{OR}}^P$, so $f\in P$. Letting $\xi<\alpha$ be a support for $f$, then we may assume that $f$ is coded by some element of $V_{\xi+1}^P=V_{\xi+1}^M$, so we may assume that $f\in M$. But then $$j(f)(a)\in j(f_x)(a_x)=x,$$ so $j(f)(a)=y$ for some $y\in x$. But then $(a,f)\approx_E(a_y,f_y)$, as desired.

Let $E$ be an extender over a structure $M$ in the language of set theory (and possibly with further symbols, which we will suppress here). We say that *Los’ criterion* holds for ${\mathrm{Ult}}(M,E)$ (where we use only functions in $M$ to form the ultrapower) iff for all $n<{\omega}$, for all $f_1,\ldots,f_n\in M$, for all $a\in\left<{\mathrm{spt}}(E)\right>^{<{\omega}}$, and all formulas $\varphi$, if there are $E_a$-measure one many $u\in\left<M\right>^{<{\omega}}$ such that $$M{\models}\exists y\varphi(f_1(u),\ldots,f_n(u),y)$$ then there is $b\in\left<{\mathrm{spt}}(E)\right>^{<{\omega}}$ and $g\in M$ such that $a{\subseteq}b$ and for $E_b$-measure one many $v$, we have $$M{\models}\varphi(f_1^{ab}(v),\ldots,f_n^{ab}(v),g(v)).\qedhere$$

\[tm:generalized\_Los\] Let $M$ be a model of ${\mathsf{ZF}}$ and let $E$ be an $M$-extender. Suppose that Los’ requirement holds for ${\mathrm{Ult}}(M,E)$. Then for all $n<{\omega}$, all $f_1,\ldots,f_n\in M$, all $a_1,\ldots,a_n\in\left<{\mathrm{spt}}(E)\right>^{<{\omega}}$ and all formulas $\varphi$, letting $a\in\left<{\mathrm{spt}}(E)\right>^{<{\omega}}$ be such that $a_i{\subseteq}a$ for each $i$, then $${\mathrm{Ult}}(M,E){\models}\varphi([a_1,f_1],\ldots,[a_n,f_n])$$ iff there are $E_a$-measure one many $v\in\left<M\right>^{<{\omega}}$ such that $$M{\models}\varphi(f_1^{a_1b}(v),\ldots,f_n^{a_nb}(v)).$$ Therefore, the ultrapower embedding $i^M_E:M\to{\mathrm{Ult}}(M,E)$ (defined as usual) is $\in$-cofinal and elementary.

This is basically the usual induction to prove Los’ theorem under ${\mathsf{AC}}$, except that we use Los’ requirement instead of appealing to ${\mathsf{AC}}$. One difference, however, is that we need to allow enlarging $a$ to $b$ in order to find an element $[b,g]$ of the ultrapower witnessing the a statement; in the usual proof of Los’ theorem, one can take $a=b$.

Definability of the class of critical points {#sec:crit_def}
============================================

Under ${\mathsf{ZFC}}$ (or better, ${\mathsf{ZFC}}_2$), the fact that $\kappa={\mathrm{cr}}(j)$ for some elementary $j:V\to M$ with $M$ transitive, is equivalent to the measurability of $\kappa$. Therefore this “criticality” of $\kappa$ is first-order. We make a brief digression to consider this question under ${\mathsf{ZF}}$ (really, ${\mathsf{ZF}}_2$).

Assume ${\mathsf{ZF}}_2$. An ordinal $\kappa$ is *critical* iff there is an elementary $j:V\to M$ with ${\mathrm{cr}}(j)=\kappa$, where $M{\subseteq}V$ is transitive.

\[lem:critical\_inaccessible\] Assume ${\mathsf{ZF}}_2$. Let $\kappa$ be critical. Then $\kappa$ is inaccessible.

Suppose not and let $\alpha<\kappa$ and $f:V_\alpha\to\kappa$ be cofinal. Let $j:V\to M$ be elementary with ${\mathrm{cr}}(j)=\kappa$. Then $j(f)=j{\circ}f=f$, although by elementarity, $j(f):V_\alpha\to j(\kappa)$ is cofinal, a contradiction.

We do not know whether ${\mathsf{ZF}}_2$ proves that criticality is a first-order property. But we will show that it is under ${\mathsf{ZF}}_2$+“There is a proper class of weak Löwenheim-Skolem cardinals”. Recall this notion from [@usuba_ls Definition 4]:

Work in ${\mathsf{ZF}}$. An uncountable cardinal $\kappa$ is *weak Löwenheim-Skolem* (*wLS*) if for every $\gamma<\kappa$ and $\alpha\in[\kappa,{\mathrm{OR}})$ and $x\in V_\alpha$, there is an elementary $X{\preccurlyeq}V_\alpha$ such that (i) $V_\gamma{\subseteq}X$, (ii) $x\in X$ and (iii) the transitive collapse of $X$ is in $V_\kappa$.

Usuba also defines *Löwenheim-Skolem* (LS) cardinals, which is at least superficially stronger. As Usuba mentions in [@usuba_ls], ${\mathsf{ZFC}}$ proves that there is a proper class of LS cardinals, and that (a result of Woodin is that) under just ${\mathsf{ZF}}$, every supercompact cardinal is an LS cardinal. Thus, assuming ${\mathsf{ZF}}$ and that there is a proper class of Reinhardt cardinals, then there is a proper class of LS cardinals, and hence wLS cardinals.

The following is clear:

\[lem:wLS\_cards\_closed\] Assume ${\mathsf{ZF}}$. Then:

1.  The class of wLS cardinals is closed.

2.  Suppose there is a proper class of wLS cardinals and let $\gamma\in{\mathrm{OR}}$ be regular. Then there is a proper class of wLS cardinals $\delta$ such that $\operatorname{cof}(\delta)=\gamma$.

In the following theorem, assuming ${\mathsf{ZF}}_2$, say that $(\kappa,\delta,N,j)$ is a *criticality pre-witness* iff $\delta$ is a weak Löwenheim-Skolem cardinal, $N$ is transitive, $j:V_\delta\to N$ is $\in$-cofinal and $\Sigma_1$-elementary and ${\mathrm{cr}}(j)=\kappa$.

Of course under ${\mathsf{ZFC}}_2$, criticality is equivalent to measurability, and has a first-order formulation. We now generalize this result:

\[tm:crit\_def\]Assume ${\mathsf{ZF}}_2$ and that there is a proper class of wLS cardinals. Let $\kappa\in{\mathrm{OR}}$. Then the following are equivalent:

1.  $\kappa$ is critical

2.  there is a criticality pre-witness $(\kappa,\delta,N,j)$ with $\operatorname{cof}(\delta)>{\omega}$,

3.  there is a criticality pre-witness $(\kappa,\delta,N,j)$ such that ${\mathrm{Ult}}(V,E_j)$ is wellfounded, where $E_j$ is the $V_\delta$-extender derived from $j$.

In particular, criticality is first-order definable over $V$. Moreover, for any criticality pre-witness $(\kappa,\delta,N,j)$, if $\operatorname{cof}(\delta)>{\omega}$ then ${\mathrm{Ult}}(V,E_j)$ is wellfounded.

Suppose first that $\kappa$ is critical, and let $k:V\to M$ be elementary with ${\mathrm{cr}}(k)=\kappa$. By Lemma \[lem:critical\_inaccessible\], $\kappa$ is regular. So by hypothesis and Lemma \[lem:wLS\_cards\_closed\], we can fix a Löwenheim-Skolem cardinal $\delta>\kappa$ with $\operatorname{cof}(\delta)>{\omega}$ (in fact $\operatorname{cof}(\delta)=\kappa$ is possible). Let $\xi=\sup k``\delta$ and $N=V_{\xi}^M$ and $j=k{\!\upharpoonright\!}V_\delta$. Then $(\kappa,\delta,N,j)$ is a criticality pre-witness with $\operatorname{cof}(\delta)>{\omega}$. We will show below that it follows that ${\mathrm{Ult}}(V,E_j)$ is wellfounded, but here it is easier. Note that we have not yet proved that Los’ theorem holds, or even that ${\mathrm{Ult}}(V,E_j)$ is extensional. Nevertheless, we can define $$\ell:{\mathrm{Ult}}(V,E_j)\to M,$$ $$\ell([a,f]^V_{E_j})=k(f)(a),$$ which, directly from the definition of “$\in$” and “$=$” in the ultrapower, is $\in$-preserving and $=$-preserving. But $M$ is transitive, so ${\mathrm{Ult}}(V,E_j)$ is wellfounded.

Now suppose that $(\kappa,\delta,N,j)$ is any criticality pre-witness such that either $U={\mathrm{Ult}}(V,E)$ is wellfounded, where $E=E_j$, or $\operatorname{cof}(\delta)>{\omega}$. We will show that anyway, $U$ is wellfounded (which we then take as transitive). Given this, it suffices to see that Los’ criterion holds for this ultrapower, because then we have the ultrapower map $k:V\to U$, which is elementary by Generalized Los’ Theorem, and $j{\subseteq}k$, so ${\mathrm{cr}}(k)=\kappa$. So:

Los’ criterion holds for $U={\mathrm{Ult}}(V,E)$.

Let $\gamma<\delta$ and $a\in \left<j(V_\gamma)\right>^{<{\omega}}$ and $f:\left<V_\gamma\right>^{<{\omega}}\to V$ and let $\varphi$ be a formula and suppose that for $E_a$-measure one many $u\in\left<V_\gamma\right>^{<{\omega}}$, there is $y$ such that $\varphi(f(u),y)$.

Let $n<{\omega}$ be large and $\alpha\in{\mathrm{OR}}$ be large and with $V_\alpha{\preccurlyeq}_n V$. Applying weak Löwenheim-Skolemness at $\delta$, let $X{\preccurlyeq}V_\alpha$ with $$V_\gamma\cup\{\gamma,f,\delta,j,N,E\}{\subseteq}X$$ and such that the transitive collapse $\bar{X}$ of $X$ is in $V_\delta$. Let $\pi:\bar{X}\to X$ be the uncollapse map. Let $\pi(\bar{f})=f$, etc.

By the elementarity, for each $u\in\left<V_\gamma\right>^{<{\omega}}$, and each $v\in\bar{X}$, we have that $\bar{X}{\models}\varphi(\bar{f}(u),v)$ iff $V{\models}\varphi(f(u),\pi(v))$.

Now consider $j(\bar{X})\in N$. We may fix $y\in\left<j(\bar{X})\right>^{<{\omega}}$ such that $j(\bar{X}){\models}\varphi(j(\bar{f})(a),y)$. Let $\xi<\kappa$ be such that $\bar{X}\in V_\xi$ and let $b\in\left<j(V_\xi)\right>^{<{\omega}}$ with $(a,y)$ coded into $b$. Then for $E_b$-measure one many $w\in\left<V_\xi\right>^{<{\omega}}$, letting $(a^w,y^w)$ be those elements corresponding to $(a,y)$, we have that $a^w\in\left<V_\gamma\right>^{<{\omega}}$ and $y^w\in\left<\bar{X}\right>^{<{\omega}}$ and $\bar{X}{\models}\varphi(\bar{f}(a^w),y^w)$, and hence $V{\models}\varphi(f(a^w),\pi(y^w))$.

So define $g:\left<V_\xi\right>^{<{\omega}}\to V$ by setting $g(w)=\pi(y^w)$ for all such $w$ (and $g(w)=\emptyset$ otherwise). Then for $E_b$-measure one many $w\in\left<V_\xi\right>^{<{\omega}}$, we have $V{\models}\varphi(f^{ab}(w),g(w))$, and therefore ${\mathrm{Ult}}(V,E){\models}\exists y\varphi([b,f^{ab}],y)$, as desired.

$U$ is wellfounded.

Suppose not. So $\operatorname{cof}(\delta)>{\omega}$. For limit ordinals $\xi<\delta$, let $E_\xi$ be the extender derived from $j{\!\upharpoonright\!}V_\xi$. Let $U_\xi={\mathrm{Ult}}(V,E_\xi)$ and $k_\xi:U_\xi\to{\mathrm{Ult}}(V,E)$ the natural factor map and $k_{\xi\zeta}:U_\xi\to U_\zeta$ likewise. We have not verified Los’ criterion for these partial ultrapowers, so we do not claim elementarity of the maps; nor do we claim that $U_\xi$ is extensional. But note that $k_\xi$ and $k_{\xi\zeta}$ are well-defined and respect “$\in$” and “$=$” (that is, in the sense of the ultrapowers, even if they fail extensionality), and commute; that is, $k_{\xi\beta}=k_{\zeta\beta}{\circ}k_{\xi\zeta}$.

Let $\xi\leq\delta$ be a limit. If ${\mathrm{Ult}}(V,E_\xi)$ is illfounded then ${\mathrm{OR}}^{{\mathrm{Ult}}(V,E_\xi)}$ is illfounded; that is, ${\mathrm{Ult}}({\mathrm{OR}},E_\xi)$ is illfounded. For, for each function $f$ used in forming $U_\xi$ (that is, $f:\left<V_\xi\right>^{<{\omega}}\to V$), let $f':\left<V_\xi\right>^{<{\omega}}\to{\mathrm{OR}}$ be $f'(u)=\operatorname{rank}(f(u))$. Then note that if $[a,f]^{V}_{E_\xi}\in[b,g]^V_{E_\xi}$ then $[a,f']^{V}_{E_\xi}\in[b,g']^V_{E_\xi}$. And if $f:\left<V_\xi\right>^{<{\omega}}\to{\mathrm{OR}}$ then $[a,f']^V_{E_\xi}=[a,f]^V_{E_\xi}$. This easily converts illfoundedness of ${\mathrm{Ult}}(V,E_\xi)$ to illfoundedness of its ordinals.

Now clearly each ${\mathrm{Ult}}({\mathrm{OR}},E_\xi)$ is a linear order.

We claim that there is $\xi<\delta$ such that ${\mathrm{Ult}}({\mathrm{OR}},E_\xi)$ is illfounded. For suppose not. Then ${\mathrm{Ult}}({\mathrm{OR}},E_\xi)$ is isomorphic to ${\mathrm{OR}}$. Note then that restricting $k_{\xi\zeta}$, we get a commuting system of order-preserving maps $$\ell_{\xi\zeta}:{\mathrm{OR}}\to{\mathrm{OR}}$$ (so $\ell_{\xi\zeta}$ is induced by $k_{\xi\zeta}$ in the natural way); that is, $$\ell_{\xi\beta}=\ell_{\zeta\beta}{\circ}\ell_{\xi\zeta}.$$ Clearly the direct limit of these maps is isomorphic to ${\mathrm{Ult}}({\mathrm{OR}},E_\delta)$, which by the preceding discussion is illfounded.

Now define a sequence $\left<\xi_n,\eta_n\right>_{n<{\omega}}$ of pairs of ordinals. Let $\xi_0=0$. We have that $\ell_{0\delta}$ is cofinal. So let $\eta_0$ be least such that $i_{0\delta}(\eta)$ is in the illfounded part (of ${\mathrm{Ult}}({\mathrm{OR}},E_\delta)$). Suppose we have constructed $(\xi_n,\eta_n)$, such that $i_{\xi_n\delta}(\eta_n)$ is in the illfounded part. Then note that there is a pair $(\xi,\eta)$ such that $i_{\xi_n\xi}(\eta_n)>\eta$ and $i_{\xi\delta}(\eta)$ is in the illfounded part. Let $(\xi_{n+1},\eta_{n+1})$ be the lexicographically least such pair. Let $\xi=\sup_{n<{\omega}}\xi_n$. Because $\operatorname{cof}(\delta)>{\omega}$, we have $\xi<\delta$. But the sequence just constructed exhibits that ${\mathrm{Ult}}({\mathrm{OR}},E_\xi)$ is illfounded, a contradiction.

So we have $\xi<\delta$ and ${\mathrm{Ult}}({\mathrm{OR}},E_\xi)$ is illfounded. Now let $n<{\omega}$ be large, let $\alpha\in{\mathrm{OR}}$ be large with $V_\alpha{\preccurlyeq}_nV$; hence, for some $\beta<\alpha$, we have $V_\alpha{\models}$“${\mathrm{Ult}}(V_{\beta},E_\xi)$ is illfounded”. Let $\xi'=\sup j``\xi$. Using the weak Löwenheim-Skolemness of $\delta$, let $X{\preccurlyeq}V_\alpha$ with $$V_\xi\cup V_{\xi'}^N\cup\{N,j,\xi,E_\xi,\beta\}{\subseteq}X$$ and the transitive collapse $\bar{X}$ of $X$ in $V_\delta$. So letting $\pi:\bar{X}\to X$ be the uncollapse map, we have $\pi(E_\xi)=E_\xi$, $\pi{\!\upharpoonright\!}V_\xi={\mathrm{id}}$, etc. And $\bar{X}{\models}$“${\mathrm{Ult}}(V_{\bar{\beta}},E_\xi)$ is illfounded”, where $\pi(\bar{\beta})=\beta$. As $\bar{X}$ is transitive and models enough of ${\mathsf{ZF}}$, therefore $$\bar{U}={\mathrm{Ult}}^{\bar{X}}(V_{\bar{\beta}}^{\bar{X}},E_\xi)$$ really is illfounded.

Now define $\sigma:\bar{U}\to N$ by setting $\sigma([a,f])=j(f)(a)$. (This makes sense, as $f\in\bar{X}\in V_\delta={\mathrm{dom}}(j)$.) Then note that (since $E_\xi$ is derived from $j$), $\sigma$ is $\in$-preserving. But then $N$ is illfounded, contradicting our assumption that $N$ is transitive. So $U$ is wellfounded, completing the proof.

Assume ${\mathsf{ZF}}$. Then $\kappa$ is “definably critical” (that is, witnessed by some definable-from-parameters class $k:V\to M$) iff there is a criticality pre-witness $(\kappa,\delta,N,j)$ such that $\operatorname{cof}(\delta)>{\omega}$.

Just use the proof of Theorem \[tm:crit\_def\], but now all relevant classes are definable from parameters.

No Reinhardt cardinal in $L(X)$ {#sec:no_Reinhardt_in_L(X)}
===============================

In this section, we return to Reinhardt cardinals, and will show that the theory ${\mathrm{ZFR}}+$“$V=L(X)$ for a set $X$” is inconsistent. In §\[sec:no\_Reinhardt\_in\_HOD(X)\] we will actually prove a stronger fact, by replacing $L(X)$ with ${\mathrm{HOD}}(X)$. But the proof for $L(X)$ is a straightforward application of ultrapowers by extenders, which we now have at our disposal. So we give this more direct proof first in this section.

The plan of the proof is as follows. Suppose $j:V\to V$ is elementary. We may assume that $X=V_\eta$ and $j(\eta)=\eta$ has cofinality ${\omega}$. Let $E$ be the $V_\eta$-extender derived from $j$. Let $U={\mathrm{Ult}}(V,E)$ and $k:V\to U$ the ultrapower map. It will suffice to see that $U$ is extensional, wellfounded and $k$ elementary, because then we have defined $k$ from the parameter $E$, contradicting Fact \[fact:suzuki\_no\_def\_j\].

\[tm:V=L(X)\_no\_R\] Assume $(V,j){\models}{\mathrm{ZFR}}$. Then there is no set $X$ such that $V=L(X)$.

Suppose otherwise. Then there is $\delta\in{\mathrm{OR}}$ such that $V=L(V_\delta)$. Take $\delta$ to be the least limit ordinal such that $V=L(V_\delta)$. Note then that, by elementarity, $j(\delta)=\delta$ and $$j{\!\upharpoonright\!}V_\delta:V_\delta\to V_\delta$$ is elementary and $j``\delta$ is cofinal in $\delta$.

Let $E$ be the $V_\delta$-extender derived from $j$, $U={\mathrm{Ult}}(V,E)$ and $k:V\to U$ be the ultrapower embedding. Because we are not assuming ${\mathsf{AC}}$, the usual proof of Los’ theorem does not quite go through. However, an adaptation will go through. (\*\*\*We remark that this proof was written prior to the general discussion on ultrapowers in §\[sec:ultrapowers\_in\_ZF\], so one might be able to apply that material to shorten the argument here now.) We also need to verify that $U$ is wellfounded. For these things we use that we $k$ is just a factor of $j$. Let $$W={\mathrm{Hull}}^V_{\Sigma_1}({\mathrm{rg}}(j)\cup V_\delta).$$

$W{\preccurlyeq}V$.

We have $W={\mathrm{Hull}}^V({\mathrm{rg}}(j)\cup V_\delta)$ because if $$y\in{\mathrm{Hull}}^V(\{j(x)\}\cup V_\delta)$$ then picking $\alpha\in{\mathrm{OR}}$ and $n<{\omega}$ with $n$ sufficiently large and $x\in V_\alpha{\preccurlyeq}_n V$, then $j(V_\alpha)=V_{j(\alpha)}{\preccurlyeq}_n V$, so $$y\in{\mathrm{Hull}}_{\Sigma_1}^V(\{j(x),j(V_\alpha)\}\cup V_\delta).$$

Now let $x\in W$ and $\varphi$ be a formula and suppose that $V{\models}\exists y\varphi(x,y)$. Then because $V=L(V_\delta)$, there is $z\in V_\delta$ and $\beta\in{\mathrm{OR}}$ such that $$V{\models}\exists y\exists\beta\in{\mathrm{OR}}\ [y\text{ is }\Sigma_1\text{-definable 
from }(V_\delta,z,\beta)\text{ and }\varphi(x,y)].$$ But then the least such $\beta$ is definable from $V_\delta,z,x$, and hence $\beta\in W$, and hence there is a witness $y\in W$.

$W{\cong}U$.

For this we use the usual factor function. Define $\pi:U\to V$, where for $(a,f)$ determining an an element of $U$, we have $$\pi([a,f]^V_E)=j(f)(a).$$

$\pi:U\to V$ is well-defined, elementary, ${\mathrm{rg}}(\pi)=W$.

We first show that $\pi$ is well-defined. So let $(a,f),(b,g)$ be given, with $$[a,f]^V_E=[b,g]^V_E.$$ So letting $c\in\langle V_\delta\rangle^{<{\omega}}$ (so $c$ is finite extensional) with $a\cup b{\subseteq}c$, and $\xi<\delta$ be sufficiently large, we have $$A=\{u\in\langle V_\xi\rangle^{{\mathrm{sig}}(c)}\bigm|f^{ac}(u)=g^{bc}(u)\}\in E_c.$$ That is, $c\in j(A)$. But this implies $$j(f)(a)=j(f^{ac})(c)=j(g^{bc})(c)=j(g)(b),$$ as required for well-definedness.

Analogous calculations show that $\pi$ is injective and $\in$-preseving.

Now clearly ${\mathrm{rg}}(\pi){\subseteq}W$. Let us show that $W{\subseteq}{\mathrm{rg}}(\pi)$. Let $y\in W$ and let $\varphi$ be a formula and $x\in V$ and $a\in V_\delta$ be such that $$y\text{ is the unique }z\text{ such that }\varphi(z,j(x),a).$$ By replacing $a$ with $\{a\}$, we may assume that $a$ is finite and extensional. Let $\xi<\delta$ be such that $a{\subseteq}V_{j(\xi)}$. Let $$f:\langle V_\delta\rangle^{<{\omega}}\to V$$ be the function where $f(u)=\text{ the unique }z\text{ such that }\varphi(z,x,u{\cap}V_\xi)$, if there is a unique such $z$, and $f(u)=\emptyset$ otherwise. Then note that $$\pi([a,f]^V_E)=j(f)(a)=y,$$ so $y\in{\mathrm{rg}}(\pi)$, as desired.

Therefore $\pi$ is an isomorphism between $U$ and $W$. Since ${\mathrm{rg}}(\pi)=W{\preccurlyeq}V$, it follows that $\pi:U\to V$ is elementary.

This completes the proof of the subclaim and claim.

Now that we know $U$ is wellfounded, we identify it with its transitive collapse. Since $V_\delta{\subseteq}W$, the transitive collapse of $W$ is just $L(V_\delta)=V$. But $U{\cong}W$, and so $U=V$. Recall that $k:V\to U=V$ is the ultrapower map.

$k:V\to U=V$ is fully elementary, $\pi{\circ}k=j$, and $k\neq{\mathrm{id}}$.

For all $x$, we have $k(x)=[\emptyset,c_x]^V_E$ where $c_x(u)=x$ for all $u$. So $$\pi(k(x))=j(c_x)(\emptyset)=j(x).$$ So $\pi{\circ}k=j$. Now if $V{\models}\varphi(x)$ then $V{\models}\varphi(j(x))$, so $V{\models}\varphi(\pi(k(x))$, and $\pi$ is elementary, so $\varphi{\models}\varphi(k(x))$. So $k$ is elementary. Finally, we have $k\neq{\mathrm{id}}$ since $k{\!\upharpoonright\!}V_\delta=j{\!\upharpoonright\!}V_\delta\neq{\mathrm{id}}$.

So $k:V\to V$ is a Reinhardt embedding. But $k$ is definable over $V$ from the parameter $E$, contradicting Fact \[fact:suzuki\_no\_def\_j\].

No $j:V_\delta\to V_\delta$ in $L(V_\delta)$, assuming $\operatorname{cof}(\delta)>{\omega}$
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\[rem:j\_in\_L(V\_delta)\] Here we will prove (under ${\mathsf{ZF}}$) that if $V=L(V_\delta)$ and $\operatorname{cof}(\delta)>{\omega}$ then there is no elementary $j:V_\delta\to V_\delta$. For this we will argue basically as in the previous section, starting from the $V_\delta$-extender $E$ derived from $j$.

But let us start with a (possible) example, which illustrates how things (might be) different if $\operatorname{cof}(\delta)={\omega}$. Assume ${\mathrm{ZFR}}$ and let $\lambda=\lambda_j$, and suppose $V_\lambda{\models}V={\mathrm{HOD}}$. (The author does not know whether one can arrange this starting just from ${\mathrm{ZFR}}$ or ${\mathrm{ZFR}}+$ some fragment of ${\mathsf{AC}}$.) So $V_\lambda{\models}{\mathsf{ZFC}}$. Let $\delta$ be least such that $V_\delta{\models}{\mathsf{ZFC}}+V={\mathrm{HOD}}$ and (in $V$) there is a $j:V_\delta\to V_\delta$. (In this case, $(V_\delta,j){\models}\neg{\mathsf{ZF}}_j$.)

Work in $L(V_\delta)$, where ${\mathsf{AC}}$ is true (since $V_\delta{\models}V={\mathrm{HOD}}$). Note that there is a tree $T$ searching for some $\bar{\delta}\leq\delta$ and an elementary $k:V_{\bar{\delta}}\to V_{\bar{\delta}}$. (Form it as follows: Let the nodes consist of tuples $(k_n,\bar{\delta}_n,\bar{\varepsilon}_n)$ such that $k_n:V_{\bar{\delta}_n}\to V_{\bar{\varepsilon}_n}$ is $\Sigma_n$ elementary and if $n>0$ then $V_{\bar{\delta}_{n-1}}{\preccurlyeq}_n V_{\bar{\delta}_n}$ and $V_{\bar{\varepsilon}_{n-1}}{\preccurlyeq}_n V_{\bar{\varepsilon}_n}$, and such that $k_{n-1}{\subseteq}k_n$ and $\bar{\varepsilon}_{n-1}<\bar{\delta}_n$).

By ${\mathsf{AC}}$ and because $j$ exists on the outside, there is a cofinal branch through $T$. But note this gives some elementary $k:V_{\bar{\delta}}\to V_{\bar{\delta}}$, and $V_{\bar{\delta}}{\models}{\mathsf{ZFC}}$. So in fact by minimality of $\delta$, we have $\bar{\delta}=\delta$.

So if the theory above is consistent then we can have an elementary $j:V_\delta\to V_\delta$ with $j\in L(V_\delta)$. Note that in this example, we have $\operatorname{cof}(\delta)={\omega}$ and $\delta=\lambda_j$ and $V_\delta{\models}{\mathsf{ZFC}}$. Is it possible to have $(V_\delta,j){\models}{\mathrm{ZFR}}$ with $j\in L(V_\delta)$ (hence $V_\delta\not{\models}{\mathsf{ZFC}}$)?

We now consider this situation assuming that $\operatorname{cof}(\delta)>{\omega}$.

\[lem:hulls\_under\_V=HOD(V\_delta)\] Assume ${\mathsf{ZF}}$. Suppose that $V={\mathrm{HOD}}(V_\delta)$ and $V_\delta$ is inaccessible. Then for every ordinal $\alpha>\delta$ and every $x\in V_\alpha$ and every $\beta<\delta$ there is an elementary $X{\preccurlyeq}V_\alpha$ and $\bar{\delta}<\delta$ and $\pi$ such that $x\in X$ and $X{\cap}V_\delta=V_{\bar{\delta}}$ and $\pi:V_{\bar{\delta}}\to X$ is a surjection.

Given $n<{\omega}$, we may assume that $V_\alpha{\preccurlyeq}_n V$, by increasing $\alpha$ and then subsuming the old $\alpha$ into the parameter $x$. So since $V={\mathrm{HOD}}(V_\delta)$, we may assume that $$\label{eqn:V_alpha_is_hull} 
V_\alpha={\mathrm{Hull}}_{\Sigma_2}^{V_\alpha}({\mathrm{OR}}\cup V_\delta).$$

Now define an increasing sequence of sets $\left<X_n\right>_{n<{\omega}}$ as follows; the aim is that $X=\bigcup_{n<{\omega}}X_n$ will be as desired. We will also define $\left<\xi_n,\pi_n\right>_{n<{\omega}}$ such that $\pi_n:V_{\xi_n}\to X_n$ is a surjection, and $X_n{\cap}\delta{\subseteq}\xi_n<\delta$, and get that $\bar{\delta}=\sup_{n<{\omega}}\xi_n$.

Set $X_0=V_\beta\cup\{x\}$ and $\xi_0$ least such that $X_0{\cap}\delta{\subseteq}\xi_0$ and choose an appropriate surjection $\pi_0:V_{\xi_0}\to X_0$.

Suppose we have $X_n,\xi_n,\pi_n$. Let $I$ be the set of pairs $(\varphi,\vec{x})$ such that $\varphi$ is a formula and $\vec{x}\in(V_{\xi_n}\cup X_n)^{<{\omega}}$ and $V_\alpha{\models}\exists y\varphi(\vec{x},y)$. Note that for each $(\varphi,\vec{x})\in I$, by line (\[eqn:V\_alpha\_is\_hull\]) there is $\eta<\delta$ and $z\in V_\eta$ and $\gamma<\alpha$ and $y\in V_\alpha$ such that $V_\alpha{\models}\varphi(\vec{x},y)$ and $y$ is definable over $V_\alpha$ from $(z,\gamma)$. For such a $z$, note that, in fact, there is such a $y$ definable over $V_\alpha$ from $(z,\vec{x})$, since we can minimize $\gamma$. Let $\eta_{\varphi,\vec{x}}$ be the least such $\eta$. Let $$\eta=(\xi_n+1)+\sup_{(\varphi,\vec{x})\in I}\eta_{\varphi,\vec{x}}.$$ By inaccessibility (and as we have $\pi_n$), we have $\eta<\delta$. Now let $$X_{n+1}={\mathrm{Hull}}_{\omega}^{V_\alpha}(V_\eta\cup X_n).$$ Let $\sigma:V_\eta^{<{\omega}}\to X_{n+1}$ be the natural surjection which is then induced by $\pi_n:V_{\xi_n}\to X_n$. Let $\xi_{n+1}=\eta+\sup(X_{n+1}{\cap}\delta)$. Again by inaccessibility, we have $\xi_{n+1}<\delta$. Let $\pi_{n+1}:V_{\xi_{n+1}}\to X_{n+1}$ agree with $\sigma$ on its domain, and $\pi_{n+1}(a)=\emptyset$ otherwise.

This completes the construction. Let $X=\bigcup_{n<{\omega}}X_n$. Let $\bar{\delta}=X{\cap}\delta$. Note that $\bar{\delta}=\sup_{n<{\omega}}\xi_n$ and $\bar{\delta}\in\delta$, by inaccessibility. Clearly $X{\preccurlyeq}V_\alpha$ by Vaught’s criterion, and note that $X{\cap}V_\delta=V_{\bar{\delta}}$, as desired.

Actually a better way to construct such a hull is just first take the hull $X'$ of $V_\beta\cup\{x\}$ (in the $V_\alpha$ with $\alpha$ as above), then let $\bar{\delta}=\sup(X'{\cap}\delta)$, and then let $X={\mathrm{Hull}}^{V_\alpha}(V_{\bar{\delta}}\cup\{x\})$, and note that $X{\cap}\delta=\bar{\delta}$, so we are done. This is because for each $n<{\omega}$, for each $\xi<\bar{\delta}$, there are witnesses to $\Sigma_n$ formulas about elements of $V_\xi$ and $x$ which are $\Sigma_{n+10}$-definable from elements of $V_{\xi'}$ with some $\xi'<\bar{\delta}$, because $V_\alpha={\mathrm{Hull}}_{\Sigma_2}^{V_\alpha}({\mathrm{OR}}\cup V_\delta)$, (so the $2$ here is fixed), so with a given $\Sigma_n$ formula, we can easily get a $\Sigma_{n+10}$-definable witness from $(z,\vec{x})$, if there is one which is $\Sigma_2$-definable from $(z,\gamma)$, some ordinal $\gamma$. We can definably find the sup of where enough $z$’s show up, and hence define that sup, and it gets $<\bar{\delta}$ therefore. We can also do the usual trick to see that one doesn’t define any more ordinals $\in[\bar{\delta},\delta)$ from new parameters in $V_{\bar{\delta}}$; i.e. we have already included the sup of all such ordinals, with respect to using parameters in $V_\xi$ and $\Sigma_n$ formulas, for each $\xi<\bar{\delta}$ and each $n<{\omega}$. So each of these sups are $<\bar{\delta}$.

[^4] Assume ${\mathsf{ZF}}$ and $V=L(V_\delta)$ for an inaccessbile $V_\delta$. Then there is no elementary $j:V_\delta\to V_\delta$.

Suppose $j$ is such. So $j``\delta$ is cofinal in $\delta$. Let $E$ be the $V_\delta$-extender derived from $j$. As in the proof of Theorem \[tm:V=L(X)\_no\_R\], it suffices to see that $U={\mathrm{Ult}}(V,E)$ is wellfounded and the ultrapower map $k:V\to U$ is elementary, as then $U=V$ and $k$ is definable from the parameter $E$.

\[clm:U\_wfd\] $U$ is wellfounded.

Suppose $U$ is illfounded. Because we are not assuming ${\mathsf{DC}}$, we can’t just extract a descending ${\omega}$-sequence from this. But there is $\alpha\in{\mathrm{OR}}$ such that ${\mathrm{Ult}}(V_\alpha,E)$ is illfounded. Let $\eta>\alpha$ be large enough that the ranking computations (of the elements of ${\mathrm{Ult}}(V_\alpha,E)$ in its wellfounded part) are made below $V_\eta$, with the wellfounded part of ordinal height $<\eta$. So $V_\eta$ computes its illfounded part, and note that ($*$) for every pair $[a,f]$ in its illfounded part (with $a\in\left<V_\delta\right>^{<{\omega}}$ and $f:\left<V_\delta\right>^{<{\omega}}\to 
V_\alpha$), there is $[b,g]$ also in the illfounded part such that $[b,g]\in^U[a,f]$.

Now using Lemma \[lem:hulls\_under\_V=HOD(V\_delta)\], let $V_\delta,j\in X{\preccurlyeq}V_{\eta}$ with $X{\cap}V_\delta=V_{\bar{\delta}}$ for some $\bar{\delta}<\delta$. Let $\bar{X}$ be the transitive collapse and $\pi:\bar{X}\to X$ the uncollapse. So $\pi(\bar{\delta})=\delta$ and let $\pi(\bar{E})=E$ etc. Then $\bar{X}{\models}$“${\mathrm{Ult}}(V_{\bar{\alpha}},\bar{E})$ is illfounded”, and $\bar{X}$ computes an illfounded part of (its version of) this ultrapower, and it satisfies that ($*$) above holds of its illfounded part. But note that $\bar{E}$ is just $E{\!\upharpoonright\!}V_{\bar{\delta}}$, and each of the functions involved in the ultrapower ${\mathrm{Ult}}^{\bar{X}}(V_{\bar{\alpha}}^{\bar{X}},\bar{E})$ have domain in $V_{\bar{\delta}}$, and $V_{\bar{\alpha}}^{\bar{X}}\in V_\delta$ and is a transitive set. Therefore the pairs $[a,f]$ forming ${\mathrm{Ult}}^{\bar{X}}(V_{\bar{\alpha}}^{\bar{X}},\bar{E})$ are also included in the ultrapower ${\mathrm{Ult}}(V_\delta,E)$, and therefore the latter is illfounded. But the latter is in fact wellfounded, because it factors through (and is actually equivalent to) $j$; that is, we can define $$k:{\mathrm{Ult}}(V_\delta,E)\to V_\delta$$ by setting $k([a,f])=j(f)(a)$, an isomorphism. This is a contradiction.

We now show that $k$ is elementary, by proving:

\[clm:U\_Los\] Los’ theorem holds; that is, for all formulas $\varphi$ and relevant pairs $(a_0,f_0),\ldots,(a_n,f_n)$, letting $c$ be finite extensional with $(\cup_{i\leq n}a_i){\subseteq}c$, we have $$U{\models}\varphi([a_0,f_0],\ldots,[a_n,f_n])$$ iff$$\{u\in\langle 
V_\delta\rangle^{<{\omega}}\bigm|\varphi(f_0^{a_0c}(u),\ldots,f_n^{a_nc}(u))\}\in 
E_c.$$

As usual we proceed by induction on the complexity of $\varphi$, and it is standard for atomic formulas and negations and Boolean combinations. Now consider existential quantifiers. Suppose we know that theorem holds for $\Sigma_n$ formulas, and consider a $\Sigma_{n+1}$ formula $$\varphi(u_0,\ldots,u_n)\iff\exists y\psi(u_0,\ldots,u_n,y),$$ where $\psi$ is $\Pi_n$. For simplicity we assume that $n=0$.

Suppose first that $U{\models}\exists y\psi([a,f],y)$. Let $y=[b,g]$ witness this. Let $c$ be finite extensional with $a\cup b{\subseteq}c$. Then since $[a,f]=[c,f^{ac}]$ and $[b,g]=[c,g^{bc}]$, and by induction, we have $$\{u\in\langle V_\delta\rangle^{<{\omega}}\bigm|\psi(f^{ac}(u),g^{bc}(u))\}\in 
E_c.$$ Therefore $$\{u\in\langle V_\delta\rangle^{<{\omega}}\bigm|\exists y\ \psi(f^{ac}(u),y)\}\in 
E_c.$$ But then it easily follows that $$\{v\in\langle V_\delta\rangle^{<{\omega}}\bigm|\exists y\ \psi(f(v),y)\}\in E_a.$$

Now suppose conversely that $$\{v\in\langle V_\delta\rangle^{<{\omega}}\bigm|\exists y\ \psi(f(v),y)\}\in E_a.$$ Let $\xi<\delta$ with $a{\subseteq}V_{j(\xi)}$. Let $\alpha\in{\mathrm{OR}}$ with $\delta,f\in V_\alpha{\preccurlyeq}_{n+1} V$.

There is $\zeta\in(\xi,\delta)$ with $\operatorname{cof}(\zeta)={\omega}$ and ${\mathrm{Hull}}^{V_\alpha}(V_\zeta\cup\{f,V_\delta\}){\preccurlyeq}V_\alpha$.

We get such a $\zeta$ using the inaccessibility of $\delta$ and that $$V={\mathrm{Hull}}_{\Sigma_1}^V({\mathrm{OR}}\cup V_\delta\cup\{V_\delta\}).$$ Here is more detail: Let $\zeta_0=\xi$. Given $\zeta_n$, let $\zeta_{n+1}$ be the least $\zeta'\in(\zeta_n,\delta)$ such that for every formula $\psi$ and all parameters $x\in V_{\zeta_n}$, if $$V_\alpha{\models}\exists y\ \psi(y,x,f,V_\delta)$$ then there is $y\in V_\alpha$ such that $$V_\alpha{\models}\psi(y,x,f,V_\delta)\text{ 
and }y\in{\mathrm{Hull}}_{\Sigma_1}^{V_\alpha}({\mathrm{OR}}\cup V_{\zeta'}\cup\{V_\delta\}).$$ Then $\zeta=\sup_{n<{\omega}}\zeta_n$ works.

Let $\zeta$ be as in the claim and let $H$ be the transitive collapse of the hull there. Let $\pi:H\to V_\alpha$ be the uncollapse map. so $\pi$ is fully elementary and $H\in 
V_\delta$. Write $\pi(\bar{\delta})=\delta$, $\pi(\bar{f})=f$ etc. We have $\xi<\zeta<{\mathrm{cr}}(\pi)$ and $V_\zeta{\subseteq}H$.

Now note that since $V_\alpha{\preccurlyeq}_{n+1}V$ and by the elementarity of $\pi$ and choice of $\xi$, we have $$\begin{array}{rcrcrcl} A&=&\{v\in\langle 
V_\xi\rangle^{<{\omega}}&\bigm|&H{\models}\exists y\ \varphi(\bar{f}(v),y)\}\\
&=&\{v\in\langle V_\xi\rangle^{<{\omega}}&\bigm|&\exists y\ 
\varphi(f(v),y)\}&\in& E_a.\end{array}$$ So $a\in j(A)$, so $$j(H){\models}\exists y\ \psi(j(\bar{f})(a),y).$$ Since $H={\mathrm{Hull}}^H(V_\zeta\cup\{\bar{f},\bar{V_\delta}\})$, we can fix a finite extensional $c\in V_{j(\zeta)}$ and a $\Sigma_k$ formula $\varrho$ such that $a{\subseteq}c$ and there is $y\in j(H)$ such that $$j(H){\models}\text{``}\psi(j(\bar{f})(a),y)\text{ and }y\text{ is the unique 
}z\text{ such that }\varrho(z,j(\bar{f}),j(\bar{V_\delta}),c)\text{''}.$$

Now working in $H$, define the function $\bar{g}:\langle 
V_\zeta\rangle^{<{\omega}}\to V$ by $$\bar{g}(u)=\text{ the unique }z\text{ such that 
}\varrho(z,\bar{f},\bar{V_\delta},u),$$ if there is a unique such $z$, and $\bar{g}(u)=\emptyset$ otherwise. Note that there is a set $A\in E_c$ such that for all $u\in A$ we have $H{\models}\psi(\bar{f}^{ac}(u),\bar{g}(u))$.

Let $g=\pi(\bar{g})$. Since $\pi$ is elementary and $\zeta<{\mathrm{cr}}(\pi)$, for all $u\in A$ we have $V_\alpha{\models}\psi(f^{ac}(u),g(u))$, so $V{\models}\psi(f^{ac}(u),g(u))$.

So $U{\models}\psi([c,f^{ac}],[c,g])$, so $U{\models}\psi([a,f],[c,g])$, so $U{\models}\exists y\psi([a,f],y)$, as desired.

This completes the proof of Los’ theorem (and hence the claim).

We can now prove the theorem. Recall that $U={\mathrm{Ult}}(V,E)$ is wellfounded by Claim \[clm:U\_wfd\], and since we have Los’ theorem (i.e. Claim \[clm:U\_Los\]), the ultrapower embedding $k:V\to U$ is fully elementary. In particular $U$ is extensional, so we can identify it with its transitive collapse. We have $V_\delta{\subseteq}U{\subseteq}V$, and ${\mathrm{OR}}{\subseteq}U{\models}{\mathsf{ZF}}$, so $V=L(V_\delta){\subseteq}U$. So $U=V$. We also have that $k{\!\upharpoonright\!}V_\delta=j{\!\upharpoonright\!}V_\delta\neq{\mathrm{id}}$. So $k$ is a Reinhardt embedding, but $k$ is definable from the parameter $(V_\delta,j)$, contradicting Fact \[fact:suzuki\_no\_def\_j\].

\[tm:V=L(V\_delta)\_cof(delta)&gt;om\] Assume ${\mathsf{ZF}}$ and $V=L(V_\delta)$ for some ordinal $\delta$ with $\operatorname{cof}(\delta)>{\omega}$. Then there is no $\Sigma_1$-elementary $j:V_\delta\to V_\delta$.

Let $\gamma=\operatorname{cof}(\delta)$. By Theorem \[tm:an\_iterate\_is\_elementary\], we may assume that (i) $$j:(V_\delta,A)\to(V_\delta,j(A))$$ is fully elementary for every $A{\subseteq}V_\delta$, (ii) if $\gamma<\delta$ then $j(\gamma)=\gamma$, and (iii) if $\gamma=\delta$ is regular but non-inaccessible, then $j(P)=P$, where $P={\mathrm{codcof}}(\delta)$.

Now let $E=E_j$. It suffices to see that the ultrapower ${\mathrm{Ult}}(V,E)$ satisfies Los’ criterion, and is wellfounded.

Los’ criterion holds.

We prove Los’ criterion by induction on numbers of quantifier alternations in formulas (as usual). Let $\varphi$ be a formula which is $\Sigma_k$ and let $\Omega\in{\mathrm{OR}}$ be such that ${\mathcal{J}}_\Omega(V_\delta){\preccurlyeq}_{k+1} L(V_\delta)$. Let $\alpha<\delta$ and $f:\left<V_\alpha\right>^{<{\omega}}\to V$ and $a\in V_\delta$ be such that for $E_a$-measure one many $u\in\left<V_\alpha\right>^{<{\omega}}$, we have $${\mathcal{J}}_\Omega(V_\delta){\models}\exists y\ [\varphi(f(u),y)].$$ For $u\in\left<V_\alpha\right>^{<{\omega}}$, let $\beta_u$ be the least $\beta<\delta$ such that there is $z\in V_\beta$ and $y\in{\mathcal{J}}_\Omega(V_\delta)$ such that ${\mathcal{J}}_\Omega(V_\delta){\models}\varphi(f(u),y)$ and $y$ is definable from ordinals and $(z,V_\delta)$ over ${\mathcal{J}}_\Omega(V_\delta)$. Given $\beta\leq\delta$, let $A_\beta$ be the set of all $u\in\left<V_\alpha\right>^{<{\omega}}$ such that $\beta_u\leq\beta$. Note that $A_\delta$ has $E_a$-measure one, by assumption and standard fine structure, and $$\beta_0<\beta_1\implies 
A_{\beta_0}{\subseteq}A_{\beta_1},$$ and $A_\delta=\bigcup_{\beta<\delta}A_\beta$.

We claim that there is $\beta<\delta$ such that $A_\beta$ is $E_a$-measure one. For suppose not. Suppose $\gamma<\delta$ (where $\gamma=\operatorname{cof}(\delta)$). Let $f:\gamma\to\delta$ be cofinal. Then $$A_\delta=\bigcup_{\xi<\gamma}A_{f(\xi)}.$$ The sequence $\left<A_{f(\xi)}\right>_{\xi<\gamma}\in V_\delta$. But (by our arrangements above) $j(\gamma)=\gamma$, so it follows that $$j(A_\delta)=j(\bigcup_{\xi<\gamma}A_{f(\xi)})=\bigcup_{\xi<\gamma}j(A_{f(\xi)}
).$$ But $a\in j(A_\delta)$, as $A_\delta$ is $E_a$-measure one. So there is $\xi<\gamma$ such that $a\in j(A_{f(\xi)})$, so $A_{f(\xi)}$ is $E_a$-measure one, a contradiction.

Now suppose instead that $\gamma=\delta$. If there is $\alpha<\delta$ and a surjection $\pi:V_{\alpha+1}\to\delta$ then argue much as before, using that $j(P)=P$. So suppose there is no such $(\alpha,\pi)$. Then $\delta$ is inaccessible. For suppose otherwise and let $\eta<\delta$ be least such that there is a cofinal $\pi:V_\eta\to\delta$. Then ${\mathrm{rg}}(\pi)$ has ordertype $\delta$, because otherwise $\delta$ is singular. But then we can modify $\pi$ to get a surjection $\pi':V_\eta\to\delta$, a contradiction. But then there is $\beta_0<\delta$ such that $A_\beta=A_{\beta_0}$ for all $\beta\in[\beta_0,\delta]$, so $A_{\beta_0}$ is $E_a$-measure one, as desired.

So we have $\beta<\delta$ with $A_\beta$ is $E_a$-measure one. Let $A'_\beta$ be the set of pairs $(u,z)$ such that $u\in\left<V_\alpha\right>^{<{\omega}}$ and $z\in V_\beta$ and $(u,z)$ are as above. Since $a\in j(A_\beta)$, we have some $w$ with $(a,w)\in j(A'_\beta)$. Now for $(u,z)\in A'_\beta$, let $g(u,z)$ be the least (in the natural ordering) $y$ which is definable from ordinal parameters over ${\mathcal{J}}_\Omega(V_\delta)$ from $(z,V_\delta)$ and such that ${\mathcal{J}}_\Omega(V_\delta){\models}\exists y\ [\varphi(f(u),y)]$. Then note that (roughly\*\*\*) (by induction), $${\mathrm{Ult}}(V,E){\models}\varphi([a,f],[(a,w),g]),$$ as desired.

${\mathrm{Ult}}(V,E)$ is wellfounded.

Consider the ultrapower $$(U,{<^*})={\mathrm{Ult}}(({\mathrm{OR}},\in),E)$$ (formed from pairs $[a,f]$ where $a\in\left<V_\delta\right>^{<{\omega}}$, $f\in V$ and $f:\left<V_\delta\right>^{<{\omega}}\to{\mathrm{OR}}$). Note that $(U,{<^*})$ is a linear order.

It suffices to see that $(U,{<^*})$ is wellfounded. For given $f:\left<V_\delta\right>^{<{\omega}}\to V$, let $f'(u)=\operatorname{rank}(f(u))$. Then $f':\left<V_\delta\right>^{<{\omega}}\to{\mathrm{OR}}$, and if $$[a,f]^V_E\in^{{\mathrm{Ult}}(V,E)}[a,g]^V_E$$ then $f'(u)<g'(u)$ for all $u$ such that $f(u)\in g(u)$, hence for $E_a$-measure one many $u$. Note that $(U,{<^*})$ is a linear order, and is set-like. For $\Omega\in{\mathrm{OR}}$ let $$(U_\Omega,{<^*_\Omega})={\mathrm{Ult}}((\Omega,{\in}),E),$$ formed using functions $f$ with ${\mathrm{rg}}(f){\subseteq}\Omega$. Note that $(U^*_\Omega,{<^*_\Omega})$ is just an initial segment of $(U,{<^*})$, and it suffices to see that each $(U^*_\Omega,{<^*_\Omega})$ is wellfounded. So fix $\Omega$.

Fix $\xi\in{\mathrm{OR}}$ with $\delta,\Omega<\xi$ and such that ${\mathcal{J}}_\xi(V_\delta){\preccurlyeq}_2 
L(V_\delta)$. Then ${^{(V_\delta)}}\Omega{\subseteq}{\mathcal{J}}_\xi(V_\delta)$. Recall that every element of $L(V_\delta)$ is $\Sigma_1$-definable from parameters in ${\mathrm{OR}}\cup V_\delta$; hence that the same holds of ${\mathcal{J}}_\xi(V_\delta)$; that is, $${\mathcal{J}}_\xi(V_\delta)={\mathrm{Hull}}_{\Sigma_1}^{{\mathcal{J}}_\xi(V_\delta)}(\xi\cup V_\delta).$$

Now since $\operatorname{cof}(\delta)>{\omega}$, there is a club $C{\subseteq}\delta$ of $\lambda<\delta$ with $j``V_\lambda{\subseteq}V_\lambda$ and $\operatorname{cof}(\lambda)={\omega}$, and hence, $j(\lambda)=\lambda$. Fix either $\lambda\in C$ or $\lambda=\delta$. Let $\mathscr{F}_\lambda$ be the set of functions $$f:\left<V_\delta\right>^{<{\omega}}\to\Omega$$ with $f\in{\mathrm{Hull}}_{\Sigma_1}^{{\mathcal{J}}_\xi(V_\delta)}(\xi\cup V_\lambda\cup\{j{\!\upharpoonright\!}V_\lambda\})$. Let $$O_\lambda={\mathrm{Ult}}^{\mathscr{F}_\lambda}
((\Omega,\in), E_\lambda),$$ where this ultrapower is formed using only pairs $[a,f]$ with $a\in\left<V_\lambda\right>^{<{\omega}}$ and $f\in\mathscr{F}_\lambda$. Note that $O_\lambda$ is also a linear order, and $O_\delta=(U^*_\Omega,{<^*_\Omega})$.

Given $\lambda\leq\zeta\leq\delta$, let $\pi_{\lambda\zeta}:O_\lambda\to O_\zeta$ be the natural map; that is, $$\pi_{\lambda\zeta}([a,f]^{\Omega,\mathscr{F}_\lambda}_{E_\lambda})=
[a,f]^{\Omega,\mathscr{F}_\zeta}_{E_\zeta}.$$ Clearly $\pi_{\lambda\zeta}$ is order-preserving, and for $\lambda_0\leq\lambda_1\leq\lambda_2$ we have $$\pi_{\lambda_0\lambda_2}=\pi_{\lambda_1\lambda_2}{\circ}\pi_{\lambda_0\lambda_1},$$ and $O_\delta=(U^*_\Omega,{<^*_\Omega})$ is just the direct limit of the $O_\lambda$’s under the maps $\pi_{\lambda\zeta}$ for $\lambda\leq\zeta<\delta$.

Now suppose that $(U^*_\Omega,{<^*_\Omega})$ is illfounded. Then we claim that there is $\lambda<\delta$ such that $O_\lambda$ is illfounded. For suppose not. Then we identify $O_\lambda$ with its ordertype $\in{\mathrm{OR}}$. Let $(\lambda_0,\alpha_0)$ be lexicographically least such that $\alpha_0<O_{\lambda_0}$ and $\pi_{\lambda_0\infty}(\alpha_0)$ in the illfounded part of $(U^*_\Omega,{<^*_\Omega})$. Given $(\lambda_n,\alpha_n)$, let $(\lambda_{n+1},\alpha_{n+1})$ be lexicographically least such that $\lambda_n<\lambda_{n+1}$ and $$\alpha_{n+1}<\pi_{\lambda_n\lambda_{n+1}}(\alpha_n)$$ and $\pi_{\lambda_{n+1}\infty}(\alpha_{n+1})$ is in the illfounded part of $(U^*_\Omega,{<^*_\Omega})$. Let $\lambda\sup_{n<{\omega}}\lambda_n$. Then $\lambda<\delta$, but clearly $O_\lambda$ is illfounded, a contradiction.

Fix $\lambda$ such that $O_\lambda$ is illfounded. Let $H={\mathrm{Hull}}_{\Sigma_{\omega}}^{{\mathcal{J}}_\xi(V_\delta)}(V_\lambda\cup\{j{\!\upharpoonright\!}V_\lambda,\Omega\})$. Let $J={\mathrm{Hull}}_{\Sigma_1}^{{\mathcal{J}}_\xi(V_\delta)}(\xi\cup V_\lambda\})$, so $J$ contains the pairs $[a,f]$ used in forming $O_\lambda$. We claim that the illfoundedness of $O_\lambda$ reflects into $H{\cap}J$. That is, let $\mathscr{I}^J$ be the set of pairs $[a,f]\in J$ which are used in forming $O_\lambda$ and are in its illfounded part. We claim that $H{\cap}\mathscr{I}^J\neq\emptyset$ and for every $[a,f]\in H{\cap}\mathscr{I}^J$, there is $[b,g]\in H{\cap}\mathscr{I}^J$ such that $a{\subseteq}b$ and $$[b,g]<^{O_\lambda}[a,f].$$ For $j{\!\upharpoonright\!}V_\lambda\in H$ and $\Omega\in H$ by definition, and since ${\mathcal{J}}_\xi(V_\delta){\preccurlyeq}_2 L(V_\delta)$, the fact that $O_\lambda$ is illfounded and the computation of its illfounded part (in their representation forms $[a,f]$) is absolute to ${\mathcal{J}}_\xi(V_\delta)$, and definable from the parameters $j{\!\upharpoonright\!}V_\lambda,\Omega$. So letting $z\in V_\lambda$ be such that there is $[a,f]\in\mathscr{I}^J$, with $f$ being $\Sigma_1$-definable over ${\mathcal{J}}_\xi(V_\delta)$ from ordinals and $z$, then note that by minimizing over ordinal parameters, there is in fact some such $f$ with $f$ being definable (if not $\Sigma_1$) over ${\mathcal{J}}_\xi(V_\delta)$ from $(a,z,\Omega,j{\!\upharpoonright\!}V_\lambda)$, and hence in $H$. So $H{\cap}\mathscr{I}^J\neq\emptyset$. Similarly, let $[a,f]\in H{\cap}\mathscr{I}^J$. So we can define $[a,f]$ over ${\mathcal{J}}_\xi(V_\delta)$ from a parameter of the form $(a,z,\Omega,j{\!\upharpoonright\!}V_\lambda)$, with $a,z\in V_\lambda$. Since $[a,f]\in\mathscr{I}^J$, we can find $z'\in V_\lambda$ and a pair $[b,g]\in\mathscr{I}^J$ such that $[b,g]$ is $\Sigma_1$-over-${\mathcal{J}}_\xi(V_\delta)$ from $z'$ and ordinal parameters, and then again minimizing over ordinals, and using the definability of $[a,f]$ and $\mathscr{I}^J$, we can find some such $g'$ such that $[b,g']\in\mathscr{I}^J$ and $[b,g']$ is definable over ${\mathcal{J}}_\xi(V_\delta)$ from the parameter $(b,z',a,z,\Omega,j{\!\upharpoonright\!}V_\lambda)$.

Now let $\bar{I}^J$ be the “transitive collapse” of $H{\cap}\bar{I}^J$. That is, $H{\cap}\bar{I}^J$ consists of certain pairs $[a,f]$ such that $a\in V_\lambda$ and $f:V_\lambda\to\Omega$. Let $X=\bigcup_{f\in H{\cap}\bar{I}^J}{\mathrm{rg}}(f)$ and $\bar{X}$ be the transitive collapse of $X$. Let $\pi:\bar{X}\to X$ be the uncollapse. Given $[a,f]\in H{\cap}\bar{I}^J$, let $\bar{f}:V_\lambda\to\bar{X}$ be such that $\pi{\circ}\bar{f}=f$. Since there are only $V_\lambda$-many elements of $H$, we then get a surjection $\sigma:V_\lambda\to\bar{X}$. We can therefore code $$\{[a,\bar{f}]\bigm|[a,f]\in H{\cap}\mathscr{I}^J\}$$ with some set $\widetilde{\mathscr{F}}{\subseteq}V_\lambda$ of pairs $[a,\widetilde{f}]$, and letting $\widetilde{X}$ be the induced code of $\bar{X}$, we have that $\widetilde{X},\widetilde{\mathscr{F}}\in V_\delta$, and $\widetilde{X}$ is a wellorder, and $\widetilde{f}:V_\lambda\to\widetilde{X}$ for each $[a,\widetilde{f}]\in\widetilde{\mathscr{F}}$. Also, $[a,f]<^{O_\lambda}[b,g]$ iff $$\widetilde{f}^{a,a\cup b}(u)<^{\widetilde{X}}\widetilde{g}^{b,a\cup b}(u)$$ for $E_{a\cup b}$-measure one many $u$.

So $j(\widetilde{X})$ is a wellorder, and for all $[a,\widetilde{f}],[b,\widetilde{g}]\in\widetilde{\mathscr{F}}$, $[a,f]<^{O_\lambda}[b,g]$ iff $$j(\widetilde{f})(a)<^{j(\widetilde{X})}j(\widetilde{g})(b).$$ But this then embeds $O_\lambda$ order-preserving into $j(\widetilde{X})$, so $j(\widetilde{X})$ is illfounded, a contradiction.

So we have shown that Los’ criterion holds for ${\mathrm{Ult}}(V,E)$, and it is wellfounded. Let $i_E:V\to{\mathrm{Ult}}(V,E)$ be the ultrapower map. Then by Generalized Los \[tm:generalized\_Los\], $i_E$ is fully elementary, and in particular, ${\mathrm{Ult}}(V,E)$ is extensional. Therefore we may identify ${\mathrm{Ult}}(V,E)$ with its transitive collapse. But $V_\delta^{{\mathrm{Ult}}(V,E)}=V_\delta$ by \[tm:extender\_coheres\], so $V=L(V_\delta){\subseteq}{\mathrm{Ult}}(V,E)$, so $V={\mathrm{Ult}}(V,E)$. So we have an elementary $i_E:V\to V$ which is definable over $V$ from the parameter $E$. This contradicts \[fact:suzuki\_no\_def\_j\], completing the proof.

Assuming ${\mathsf{ZF}}$, we write ${\mathcal{H}}_{{\mathrm{OR}}}=\bigcup_{\alpha\in{\mathrm{OR}}}{\mathcal{H}}_\alpha$. Equivalently, $${\mathcal{H}}_{{\mathrm{OR}}}=L[{\mathcal{P}}({<{\mathrm{OR}}})]=\bigcup_{X\in{\mathcal{P}}({<{\mathrm{OR}}})}L[X],$$ where ${\mathcal{P}}({<{\mathrm{OR}}})=\{A\bigm|\exists\alpha\in{\mathrm{OR}}\ [A{\subseteq}\alpha]\}$.

Assume ${\mathsf{ZF}}$. Note that ${\mathcal{H}}_{{\mathrm{OR}}}$ is the union of all transitive inner models $M$ of ${\mathsf{ZFC}}$. Therefore, there is a largest such $M_0$ iff ${\mathcal{H}}_{{\mathrm{OR}}}{\models}{\mathsf{ZFC}}$, in which case ${\mathcal{H}}_{{\mathrm{OR}}}=M_0$. Note also that $${\mathsf{AC}}\iff V={\mathcal{H}}_{{\mathrm{OR}}}.$$ In general, ${\mathcal{H}}_{{\mathrm{OR}}}{\models}{\mathsf{AC}}$, but we can easily have ${\mathcal{H}}_{{\mathrm{OR}}}{\models}\neg{\mathsf{ZF}}$; for example, clearly ${\mathbb R}{\subseteq}{\mathcal{H}}_{{\mathrm{OR}}}$, but if ${\mathbb R}$ is not wellorderable then ${\mathcal{H}}_{{\mathrm{OR}}}{\models}$“There is no set of all reals”. If ${\mathcal{H}}_{{\mathrm{OR}}}{\models}{\mathsf{ZF}}$ then ${\mathcal{H}}_{{\mathrm{OR}}}{\models}{\mathsf{ZFC}}$.

Assume ${\mathsf{ZF}}$ and that ${\mathcal{H}}_{\mathrm{OR}}{\models}{\mathsf{ZFC}}$. Then there is no Reinhardt embedding.

Suppose $j:V\to V$ is a Reinhardt embedding and $M={\mathcal{H}}_{{\mathrm{OR}}}{\models}{\mathsf{ZFC}}$. Then for each $\alpha\in{\mathrm{OR}}$, $V_\alpha^M=V_\alpha{\cap}M$ exists and is in $M$. Therefore $V_\alpha^M\in L[X]$ for some set $X\in{\mathcal{P}}({<{\mathrm{OR}}})$. Note that $$j{\!\upharpoonright\!}M:M\to M$$ is elementary. Now we claim that $j{\!\upharpoonright\!}M$ is amenable to $M$, and therefore $(M,j{\!\upharpoonright\!}M){\models}{\mathsf{ZFC}}$, which gives Kunen’s inconsistency. To see this, let $\alpha\in{\mathrm{OR}}$, and let $X{\subseteq}\beta\in{\mathrm{OR}}$ be such that $V_\alpha^M\in L[X]$. Then by the elementarity of $j$, $$j{\!\upharpoonright\!}L[X]:L[X]\to L[j(X)]$$ is elementary and $j(X){\subseteq}j(\beta)\in{\mathrm{OR}}$, and $V_{j(\alpha)}^M\in L[j(X)]$. But note that $$j{\!\upharpoonright\!}L_\gamma[X]\in M$$ for each $\gamma\in{\mathrm{OR}}$. But $V_\alpha^M\in L_\gamma[X]$ for some $\gamma$, so $j{\!\upharpoonright\!}V_\alpha^M\in M$, as desired.

No Reinhardt in ${\mathrm{HOD}}(X)$ {#sec:no_Reinhardt_in_HOD(X)}
===================================

As the reader familiar with inner model theory will see, the methods in this section are heavily based on analyses of ${\mathrm{HOD}}$ via direct limit systems of mice, and related methods from the study of Varsovian models. The present context actually provides a nice introduction to some of those ideas, which avoids various issues (iteration trees, fine structure, comparison).

We start with an easy observation. Assume $(V,j){\models}{\mathrm{ZFR}}$ and let $\lambda=\lambda_{j}$. Then for each $X\in V_\lambda$:

1.  There is no cofinal $f:{\omega}\to\lambda$ with $f\in{\mathrm{OD}}_X$.

2.  $j{\!\upharpoonright\!}\lambda\notin{\mathrm{OD}}_X$.

For suppose $f$ is ${\mathrm{OD}}_X$. Then there is in fact some such $f$ which is outright definable from $X$, by minimizing ordinal parameters. Taking $n<{\omega}$ large enough, then $j^n(X)=X$, so $j^n(f)=f$, so $j^n(f(k))=f(k)$ for each $k<{\omega}$. But ${\mathrm{rg}}(f)$ is unbounded in $\lambda$, and $\lambda$ is the least fixed poinit of $j^n$ which is $>{\mathrm{cr}}(j^n)$, a contradiction.

From $j{\!\upharpoonright\!}\lambda$ we can define $\left<\kappa_n\right>_{n<{\omega}}$, which we have just seen is not ${\mathrm{OD}}_X$, so $j{\!\upharpoonright\!}\lambda$ is not ${\mathrm{OD}}_X$.

Mohammad Golshani asked, on the discussion board Mathoverflow, what properties might hold of the intersection of the models $N_\alpha$ which we now define.[^5]

Assume $(V,j){\models}{\mathrm{ZFR}}$. Set $M_0=(V,j){\models}{\mathsf{ZF}}$. We define, for $\alpha\leq\beta\in{\mathrm{OR}}$, the $\alpha$th iterate $M_\alpha=(N_\alpha,j_\alpha)$, where $N_\alpha$ is a transitive proper class and $j_\alpha:N_\alpha\to N_\alpha$ a Reinhardt embedding of $N_\alpha$ (so $(N_\alpha,j_\alpha){\models}{\mathsf{ZF}}$) and $i_{\alpha\beta}:M_\alpha\to M_\beta$ is elementary (literally a class function $N_\alpha\to N_\beta$, but elementary with respect to the predicates $j_\alpha,j_\beta$), and such that $$i_{\alpha\gamma}=i_{\beta\gamma}{\circ}i_{\alpha\beta}$$ for all $\alpha\leq\beta\leq\gamma$, and $i_{\alpha\alpha}={\mathrm{id}}$, as follows. Given $M_\beta$, set $$M_{\beta+1}=(N_\beta,j_\beta(j_\beta))$$ and $i_{\alpha,\beta+1}=j_\beta{\circ}i_{\alpha\beta}$. Given $M_\beta$ for all $\beta<\eta$ where $\eta$ is a limit, set $$N_\eta={\mathrm{dir lim}}_{\alpha\leq\beta<\eta}(N_\alpha,N_\beta;i_{\alpha\beta}),$$ $$i_{\alpha\eta}=\text{ the associated direct limit map,
for }\alpha<\eta,$$ $$j_\eta=i_{\alpha\eta}(j_\alpha),$$ for any/all $\alpha<\eta$.

By elementarity, $N_\alpha$ is extensional. Assuming that $N_\alpha$ is wellfounded, we identify it with its transitive collapse.

As pointed out by Hamkins on Mathoverflow (see Footnote \[ftn:mof\]), the usual proof of linear iterability with respect to a single normal measure gives:

For all $\alpha\in{\mathrm{OR}}$, $N_\alpha$ is wellfounded, $N_\alpha$ is a transitive proper class, $j_\alpha{\subseteq}N_\alpha$ is amenable to $N_\alpha$ and is a Reinhardt embedding of $N_\alpha$, and $i_{\beta\alpha}:M_\alpha\to M_\beta$ is elementary and $\in$-cofinal.[^6]

Everything is proved by induction. We get the elementarity because $M_0=(N_0,j){\models}{\mathsf{ZF}}$, so for each $n<{\omega}$ there is a club class of $\alpha$ such that $(V_\alpha^{N_0},j{\!\upharpoonright\!}V_\alpha^{N_0}){\preccurlyeq}_n(N_0,j)$, which easily gives that $i_{01}$ is elementary with respect to $M_0,M_1$, not just $N_0,N_1$, and likewise for larger indices.

Now suppose that $\eta\in{\mathrm{OR}}$ and $N_\eta$ is illfounded, and $\eta$ is least such. Then ${\mathrm{OR}}^{N_\eta}$ is illfounded. Since $N_{\alpha+1}=N_\alpha$ by definition, $\eta$ is a limit ordinal. The construction $\left<M_\alpha\right>_{\alpha\in{\mathrm{OR}}}$ is defined Let $\lambda\in{\mathrm{OR}}$ be least such that $i_{0\eta}(\lambda)$ is in the illfounded part of ${\mathrm{OR}}^{N_\eta}$. Then there is $\alpha<\eta$ and $\lambda_1\in{\mathrm{OR}}$ such that $\lambda'=i_{0\alpha}(\lambda)>\lambda_1$ and $i_{\alpha\eta}(\lambda_1)$ is in the illfounded part of $N_\eta$. Let $\eta'=i_{0\alpha}(\eta)$. So $\eta\leq\eta'$. By the elementarity of $i_{0\alpha}$, $M_\alpha{\models}$“$\eta'$ is the least ordinal $\gamma$ such that the $N_\gamma^{M_\alpha}$”, where $N_\gamma^{M_\alpha}$ is the $\gamma$th iterate of $M_\alpha$, as computed by $M_\alpha$. But since $M_\alpha$ is wellfounded, note that $N_\gamma^{M_\alpha}$ is just $N_{\alpha+\gamma}$. So we must have $\eta'\leq\eta$, so in fact $\eta'=\eta$. Also by elementarity, $M_\alpha{\models}$“$\lambda'$ is the least ordinal such that $i^{M_\alpha}_{0\eta}(\lambda')$ is in the illfounded part of $N_\eta$”, where $i^{M_\alpha}_{0\eta}$ is the iteration map as computed by $M_\alpha$. But $i^{M_\alpha}_{0\eta}=i_{\gamma\eta}$, and we chose $\alpha$ such that $i_{\gamma\eta}(\lambda_1)$ is in the illfounded part of $N_\eta$, a contradiction.

\*\*\*Change of notation below...

From now on, ${\mathrm{ZFR}}$ uses the langugage with symbols $\in,\widetilde{j}$ (and has axioms asserting that $\widetilde{j}:V\to V$ is elementary). Working in ${\mathrm{ZFR}}$, for $\alpha,\beta\in{\mathrm{OR}}$, we write $$\widetilde{M}_\alpha=(\widetilde{N}_\alpha,\widetilde{j}_\alpha)$$ and $$\widetilde{i}_{\alpha\beta}:\widetilde{M}_\alpha\to\widetilde{M}_\beta$$ to formally refer to the $\alpha$th iterate, etc, with the formal language.

The following lemma will be very familiar to those familiar with directed limits of mice:

\[lem:alpha\*=\] Suppose $(V,j){\models}{\mathrm{ZFR}}$ and let $M_\alpha=(N_\alpha,j_\alpha)$ be the $\alpha$th iterate of $(V,j)$ and $i_{\alpha\beta}:M_\alpha\to M_\beta$ the iteration map. For $\alpha\in{\mathrm{OR}}$ let $$\alpha^*=\min_{n<{\omega}}i_{n{\omega}}(\alpha).$$ Then $\alpha^*=i_{{\omega},{\omega}+{\omega}}(\alpha)=\widetilde{i_{0{\omega}}}^{M_{\omega}}$.

Because $N_{\omega}$ is wellfounded, for each $\alpha$ there is $n_\alpha<{\omega}$ such that for all $m\in[n_\alpha,{\omega})$, we have $i_{n_\alpha m}(\alpha)=\alpha$. Therefore $$\alpha^*=i_{n_\alpha{\omega}}(\alpha)=i_{m{\omega}}(\alpha)$$ for all $m\in[n_\alpha,{\omega})$. Note that for all such $m$, $$M_m{\models}\text{``}\alpha^*=\widetilde{j}_{0{\omega}}(\alpha)\text{''.}$$ Now since $M_{\omega}$ is a direct limit, if $m$ is large enough then $\alpha\in{\mathrm{rg}}(i_{m{\omega}})$. This is a first order statement over $M_m$ about $\alpha$, and so in fact it is already true at $n_\alpha$. Let $\bar{\alpha}$ be such that $$M_{n_\alpha}{\models}\text{``}\widetilde{i}_{0{\omega}}(\bar{\alpha})=\alpha\text{''}.$$ Then we have $$i_{n_\alpha{\omega}}(\bar{\alpha},\alpha)=(\alpha,\alpha^*).$$ Therefore $$M_{\omega}{\models}\text{``}\widetilde{i_{0{\omega}}}(\alpha)=\alpha^*\text{''}.$$ Therefore $i_{{\omega},{\omega}+{\omega}}(\alpha)=\alpha^*$, as desired.

Suppose $(V,j){\models}{\mathrm{ZFR}}$ and let $M_\alpha$ be the $\alpha$th iterate of $(V,j)$. Then ${\mathrm{HOD}}{\subseteq}N_{\omega}$. Moreover, $\lambda=\kappa_{{\omega},j}={\mathrm{cr}}(j_{\omega})$ is Mahlo in ${\mathrm{HOD}}$.

Let $X{\subseteq}\gamma\in{\mathrm{OR}}$ with $X\in{\mathrm{HOD}}$. Let $\varphi$ be a formula and $\eta\in{\mathrm{OR}}$ such that for all $\alpha<\gamma$, we have $$\alpha\in X\iff\varphi(\eta,\alpha).$$ Note that for $m\in[n_\eta,{\omega})$ we have $i_{n_\eta m}(X)=X$, since $i_{n_\eta m}(\eta)=\eta$ and $N_{n_\eta}=V=N_m$. Let $X^*=i_{n_\eta{\omega}}(X)=i_{m{\omega}}(X)$ for all $m\in[n_\eta,{\omega})$. So $X^*\in M_{\omega}$, and note that for $\alpha\in{\mathrm{OR}}$ we have $$\alpha\in X\iff\alpha^*\in X^*\iff 
M_{\omega}{\models}\text{``}\widetilde{i}_{0{\omega}}(\alpha)\in X^*\text{''},$$ where we have used Lemma \[lem:alpha\*=\]. Therefore $X\in M_{\omega}$, as desired.

Now since $\lambda={\mathrm{cr}}(j_{\omega})$, $\lambda$ is inaccessible in $M_{\omega}$, and since ${\mathrm{HOD}}{\subseteq}M_{\omega}$, therefore inaccessible in ${\mathrm{HOD}}$. Now let $C\in{\mathcal{P}}(\lambda){\cap}{\mathrm{HOD}}$ be club in $\lambda$. Since $C\in M_{\omega}$, there is $n<{\omega}$ such that $C=i_{n{\omega}}(C{\cap}\kappa_n)$ where $\kappa_n={\mathrm{cr}}(i_{n{\omega}})$. But then $C{\cap}\kappa_n$ is club in $\kappa_n$, so $\kappa_n\in i_{n{\omega}}(C{\cap}\kappa_n)=C$, and $\kappa_n$ is regular in ${\mathrm{HOD}}$, as desired.

Is $\lambda=\kappa_{{\omega},j}$ weakly compact in ${\mathrm{HOD}}$? Measurable in ${\mathrm{HOD}}$?

\[tm:every\_om\] Let $(V,j){\models}{\mathrm{ZFR}}$. Then every set in $V$ is set-generic over $M_{\omega}^{(V,j)}$.

We will show directly that for each $\beta\in{\mathrm{OR}}$ and each $X{\subseteq}V_\beta^{M_{\omega}}$, with $X\in V$, we can add $X$ set-generically to $M_{\omega}$. But this suffices, because letting $\alpha\in{\mathrm{OR}}$, we can apply this to $$X=i_{0{\omega}}``V_\alpha,$$ and clearly then $V_\alpha^{M_{\omega}[G]}=V_\alpha$ (and also, this adds $j{\!\upharpoonright\!}V_\alpha$ too).

The forcing ${\mathbb P}$ we use is the natural adaptation of Schindler’s forcing (to appear in [@vm2]) to the current setting (and Schindler’s forcing is related to the forcings of Vopenka, Bukowsky and Woodin’s extender algebra).

Now work in $M_{\omega}$. Let ${\mathcal{L}}$ be the class of all infinitary Boolean formulas $\varphi$ in propositional symbols $P_x$, for each $x\in V_\beta$. That is, ${\mathcal{L}}$ is the minimal class generated by the following rules:

1.  For each $x\in V_\beta$, we have a corresponding propositional symbol $P_x\in{\mathcal{L}}$ (with $x\neq y\implies P_x\neq P_y$).

2.  If $\varphi\in{\mathcal{L}}$ then $\neg\varphi\in{\mathcal{L}}$.

3.  If $A$ is a set and $A{\subseteq}{\mathcal{L}}$ then $\bigvee A\in{\mathcal{L}}$ and $\bigwedge A\in{\mathcal{L}}$.

Working for a moment in a possibly larger universe, given $\varphi\in{\mathcal{L}}$ and a set $X{\subseteq}V_\beta^{M_{\omega}}$, we define satisfaction $X{\models}\varphi$ recursively in the obvious manner:

1.  $X{\models}P_x\iff x\in X$,

2.  $X{\models}\neg\varphi\iff\neg (X{\models}\varphi)$,

3.  $X{\models}\bigvee A\iff$ $X{\models}\varphi$ for some $\varphi\in A$,

4.  $X{\models}\bigwedge A\iff$ $X{\models}\varphi$ for all $\varphi\in A$.

Now recall that $\beta=\sup i_{0{\omega}}``\alpha$, and work again in $M_{\omega}$. Write $x^*=\widetilde{i}_{0{\omega}}^{M_{\omega}}(x)$. Let ${\mathcal{L}}'$ be the class of formulas $\varphi\in{\mathcal{L}}$ such that there is $X{\subseteq}V_{\beta^*}$ such that $X{\models}\varphi^*$. Given $\varphi\in{\mathcal{L}}'$, let $$\mathscr{E}_\varphi=\{X{\subseteq}V_{\beta^*}\bigm|X{\models}\varphi^*\}.$$ So $\mathscr{E}_\varphi\neq\emptyset$ by definition. Given $\varphi,\psi\in{\mathcal{L}}'$, set $\varphi\equiv\psi$ iff $\mathscr{E}_\varphi=\mathscr{E}_\psi$. So there are only set-many (in fact at most $V_{\beta^*+2}$-many) equivalence classes. Let ${\mathbb P}$ be the partial order whose conditions are the equivalence classes $[\varphi]$, and with $$[\varphi]\leq[\psi]\iff\mathscr{E}_{\varphi}{\subseteq}\mathscr{E}_{\psi}
.$$ Clearly this is a partial order (and note that it is isomorphic to a partial order ${\subseteq}V_{\beta^*+2}$).

Now work in $V$ and let $X{\subseteq}V_\beta$. Let $$G_X=\{[\varphi]\in{\mathbb P}\bigm|\varphi\in{\mathcal{L}}'\text{ and }X{\models}\varphi\}.$$ Here if $\varphi,\psi\in{\mathcal{L}}'$ and $\varphi\equiv\psi$ then $X{\models}\varphi\Leftrightarrow X{\models}\psi$. For suppose that $X{\models}\varphi\wedge\neg\psi$. Let $n<{\omega}$ be large enough that $i_{nm}(\varphi\wedge\neg\psi)=\varphi\wedge\neg\psi$ and $i_{nm}(\beta)=\beta$ for all $m\in[n,{\omega})$. Then note that $$V{\models}\text{``There is }X'{\subseteq}V_\beta\text{ such that 
}X'{\models}\varphi\wedge\neg\psi\text{''}$$ (as witnessed by $X$). Therefore, applying $i_{n{\omega}}$, we have $$M_{\omega}{\models}\text{``There is }X'{\subseteq}V_{\beta^*}\text{ such that }
 X'{\models}(\varphi\wedge\neg\psi)^*\text{''}.$$ But then for any such $X'$, we have $X'{\models}\varphi^*$ and $\neg(X'{\models}\psi^*)$ (as $(\varphi\wedge\neg\psi)^*=\varphi^*\wedge\neg(\psi^*)$), so $\mathscr{E}_\varphi\neq\mathscr{E}_\psi$, a contradiction.

Now we claim that $G_X$ is $(M_{\omega},{\mathbb P})$-generic; and then it is easy to see that $M_{\omega}[G_X]=M_{\omega}[X]$. For $G_X$ is easily a filter, so we just need to verify genericity. So let $D\in{\mathcal{P}}({\mathbb P}){\cap}M_{\omega}$ be dense. We must see that $G_X{\cap}D\neq\emptyset$. Note that since $D$ is a set, for each $\xi\in{\mathrm{OR}}$, $M_{\omega}$ has the set $$E=\{\varphi\in{\mathcal{L}}'{\cap}V_\xi\bigm|[\varphi]\in D\},$$ and taking $\xi$ large enough, we have $$D=\{[\varphi]\bigm|\varphi\in E\}.$$ Now $\bigvee E\in{\mathcal{L}}$. We must see that $X{\models}\bigvee E$, because then $X{\models}\varphi$ for some $\varphi\in E$, and therefore $G_X{\cap}D\neq\emptyset$.

So suppose that $X{\models}\neg\bigvee E$. Let $n<{\omega}$ be large enough that $i_{nm}(E)=E$ and $i_{nm}(\beta)=\beta$ for all $m\in[n,{\omega})$. Then note that $$V{\models}\text{``There is }X'{\subseteq}V_\beta\text{ such that }X'{\models}\neg\bigvee 
E\text{''}$$ (as witnessed by $X$). Therefore $$M_{\omega}{\models}\text{``There is }X'{\subseteq}V_{\beta^*}\text{ such that 
}X'{\models}(\neg\bigvee E)^*\text{''}.$$ So $\neg\bigvee E\in{\mathcal{L}}'$. By the density of $D$, there is $\varphi\in E$ such that $[\varphi]$ is compatible with $[\neg\bigvee E]$. So let $\psi\in{\mathcal{L}}'$ with $$[\psi]\leq[\varphi]\text{ and }[\psi]\leq[\neg\bigvee E].$$ This says that $$\emptyset\neq\mathscr{E}_\psi{\subseteq}\mathscr{E}_\varphi{\cap}\mathscr{E}_{
\neg\bigvee E}.$$ So let $Y\in M_{\omega}$, with $Y{\subseteq}V_{\beta^*}$, witness that $\mathscr{E}_\varphi{\cap}\mathscr{E}_{\neg\bigvee E}\neq\emptyset$. Then $$Y{\models}\varphi^*\text{ and }
 Y{\models}(\neg\bigvee E)^*,$$ but note that $$(\neg\bigvee 
E)^*=\neg\bigvee(E^*)\equiv\bigwedge\{\neg\varrho\bigm|\varrho\in E^*\}$$ and $\varphi\in E$, so $\varphi^*\in E^*$, so $$Y{\models}\varphi^*\text{ and }Y{\models}\neg\varphi^*,$$ a contradiction.

Suppose $(V,j){\models}{\mathrm{ZFR}}$. Let $\alpha<{\omega}^2$. Then every set in $V$ is set-generic over $M_\alpha$.

We have this for $\alpha<{\omega}+{\omega}$, by Theorem \[tm:every\_om\]. Consider then $\alpha={\omega}+{\omega}$. By Theorem \[tm:every\_om\], applied *in* $M_{\omega}$, we get that every set in $M_{\omega}$ is set-generic over $M_{{\omega}+{\omega}}$. Therefore given any $\alpha\in{\mathrm{OR}}$, we can find a set-generic extension $M_{{\omega}+{\omega}}[G]$ of $M_{{\omega}+{\omega}}$ such that $V_\alpha^{M_{{\omega}+{\omega}}[G]}=V_\alpha^{M_{\omega}}$. But then for any set $X\in V$, by taking $\alpha$ high enough, we get that $X$ is set-generic over $M_{{\omega}+{\omega}}[G]$. So we have added $X$ by a 2-step iteration over $M_{{\omega}+{\omega}}$, which suffices. Clearly this generalizes to all $\alpha<{\omega}+{\omega}$.

Suppose $(V,j){\models}{\mathrm{ZFR}}$ and let $\alpha$ be a limit ordinal. Say that $x$ is *$\alpha$-eventually stable* (*$\alpha$-es*) iff $x\in\bigcap_{\xi<\alpha}M_\xi$ and there is $\xi<\alpha$ such that $i_{\xi\gamma}(x)=x\text{ for all }\gamma\in[\xi,\alpha)$. Say that $x$ is *hereditarily $\alpha$-eventually stable* (*$\alpha$-hes*) iff every $y\in{\mathrm{trancl}}(\{x\})$ is $\alpha$-es. For $\alpha$-es $x$, let $x^*=\lim_{\xi<\alpha}i_{\xi\alpha}(x)$. Note here that $i_{\xi\alpha}(x)=i_{\gamma\alpha}(x)$ whenever $\xi\leq\gamma<\alpha$ and $\xi$ witnesses the $\alpha$-eventual stability of $x$.

Given an ordinal $\alpha>0$, the *limit-ordertype* ${\mathrm{lot}}(\alpha)$ of $\alpha$ is the least $\chi$ such that $\alpha=\xi+\chi$ for some $\xi<\alpha$.

Note that (i) $\alpha$ is a successor iff ${\mathrm{lot}}(\alpha)=1$ iff ${\mathrm{lot}}(\alpha)$ is a successor; (ii) ${\mathrm{lot}}(\alpha)={\mathrm{lot}}({\mathrm{lot}}(\alpha))$; (iii) $\alpha=\xi+{\mathrm{lot}}(\alpha)$. for all sufficiently large $\xi<\alpha$.

\[lem:star\_general\]Suppose $(V,j){\models}{\mathrm{ZFR}}$. Let $\alpha$ be a limit ordinal. Then:

1.  $x\in M_\alpha$ iff $x$ is $\alpha$-hes (so all ordinals are $\alpha$-hes).

2.  if $x\in M_\alpha$ then $x^*=i_{\alpha,\alpha+\eta^*}(x)=\widetilde{i}^{M_\alpha}_{0\eta^*}(x)$ where $\eta={\mathrm{lot}}(\alpha)$.

First note that all ordinals all $\alpha$-es, since $M_\alpha$ is wellfounded.

Now let $x\in M_\alpha$; we show that $x$ is $\alpha$-hes. We have $M_\alpha{\subseteq}M_\xi$ for each $\xi<\alpha$, because in fact $M_\xi=\widetilde{M}_\beta^{M_\alpha}$ where $\alpha+\beta=\xi$. Assume by induction that every $y\in{\mathrm{trancl}}(x)$ is $\alpha$-es.

We just need to see that $x$ is $\alpha$-es. Let $\xi<\alpha$ be such that $x\in{\mathrm{rg}}(i_{\xi\alpha})$ and $\xi+\eta=\alpha$ where $\eta={\mathrm{lot}}(\alpha)$, and $i_{\xi\gamma}(\eta)=\eta$ for all $\gamma\in[\xi,\alpha)$. We claim that $i_{\xi\gamma}(x)=x$ for all $\gamma\in[\xi,\alpha)$. For $$x=i_{\xi\alpha}(\bar{x})=\widetilde{i}^{M_\xi}_{0\eta}(\bar{x}),$$ so $$i_{\xi\gamma}(x)=i_{\xi\gamma}(\widetilde{i}^{M_\xi}_{0\eta})(i_{\xi\gamma}
(\bar{x}))=\widetilde{i}^{M_\gamma}_{0i_{\xi\gamma}(\eta)}(i_{\xi\gamma}(\bar{x}
)),$$ but by our choices, $i_{\xi\gamma}(\eta)=\eta$ and $\gamma+\eta=\alpha$, so this is $$=\widetilde{i}^{M_\gamma}_{0\eta}(i_{\xi\gamma}(\bar{x}))=
i_{\gamma\alpha}(i_{\xi\gamma}(\bar{x}))=
i_{\xi\alpha}(\bar{x})=x,$$ as desired.

Now let $x\in M_\alpha$ and $\eta={\mathrm{lot}}(\alpha)$. We must see that $x^*=i_{\alpha,\alpha+\eta^*}(x)$. Let $\xi<\alpha$ be such that (i) $\xi+\eta=\alpha$, (ii) $x\in{\mathrm{rg}}(i_{\xi\alpha})$, and (iii) $i_{\xi\gamma}(x,\eta)=(x,\eta)$ for all $\gamma\in[\xi,\alpha)$. Then $(x^*,\eta^*)=i_{\xi\alpha}(x,\eta)$; let $\bar{x}$ be such that $i_{\xi\alpha}(\bar{x})=x$. Since $M_\xi{\models}$“$x=\widetilde{i}_{0\eta}(\bar{x})$”, applying $i_{\xi\alpha}$, we get $$M_\alpha{\models}\text{``}x^*=\widetilde{i}_{0\eta^*}(x)\text{''},$$ so $i_{\alpha,\alpha+\eta^*}(x)=x^*$, as desired.

\[tm:every\_set\_generic\_over\_N\_tau\] Suppose $(V,j){\models}{\mathrm{ZFR}}$ and let $\tau\in{\mathrm{OR}}$. Then every set in $V$ is set-generic over $M_\tau$.

By induction on $\tau$. Successor steps are trivial, so let $\tau$ be a limit. Let $\alpha\in{\mathrm{OR}}$ and $\beta=\sup i_{0\tau}``\alpha$. Let $X=i_{0\tau}``V_\alpha$. It suffices to see that $X$ is set-generic over $M_\tau$. We will just use the fact that $X\in V$ and $X{\subseteq}V_{\beta}^{M_\tau}$.

Since $X\in V$, $X$ is set-generic over $M_\xi$ for each $\xi<\tau$. Note then that we can fix $\delta\in{\mathrm{OR}}$ for each $\xi<\tau$, there is an $(M_\xi,{\mathrm{Col}}({\omega},V_\delta^{M_\xi}))$-generic $G$ such that $X\in M_\xi[G]$.

Let $\chi={\mathrm{lot}}(\tau)$. For $x\in M_\tau$, let $x^*=\lim_{\xi<\tau}i_{\xi\tau}(x)$, so by Lemma \[lem:star\_general\], we have $x^*=\widetilde{i}^{M_\tau}_{0\chi^*}(x)$.

Work in $M_\tau$. Let ${\mathcal{L}}$ be the language formed as before (with a propositional symbol $P_x$ for each $x\in V_\beta$). Let ${\mathbb Q}={\mathrm{Col}}({\omega},V_{\delta^*})$. For $\varphi\in{\mathcal{L}}$ and $G$ a $(V,{\mathbb Q})$-generic, let $\mathscr{E}_{\varphi}$ be the set of all $X\in{\mathcal{P}}(V_{\beta^*}^{\widetilde{M}_{\chi^*}}){\cap}V[G]$ such that $X{\models}\varphi^*$. Let ${\mathcal{L}}'$ be the class of all $\varphi\in{\mathcal{L}}$ such that ${\mathbb Q}{\dststile{}{}}\mathscr{E}_\varphi\neq\emptyset$. Given $\varphi,\psi\in{\mathcal{L}}'$, let $\varphi\equiv\psi$ iff ${\mathbb Q}{\dststile{}{}}\mathscr{E}_\varphi=\mathscr{E}_\psi$. Let $[\varphi]$ be the equivalence class of $\varphi\in{\mathcal{L}}'$. Let ${\mathbb P}$ be the partial order whose conditions are the equivalence classes $[\varphi]$, and with $[\varphi]\leq[\psi]$ iff ${\mathbb Q}{\dststile{}{}}\mathscr{E}_\varphi{\subseteq}\mathscr{E}_\psi$.

Now work in $V$. Let $G_X=\{\varphi\in{\mathcal{L}}'\bigm|X{\models}\varphi\}$. We claim that $G$ is $(M_\tau,{\mathbb P})$-generic; it follows that $M_\tau[G_X]=M_\tau[X]$. This is proven like before. Let $D\in{\mathcal{P}}({\mathbb P}){\cap}M_\tau$ be dense. Let $E\in M_\tau$ be a such that $D=\{[\varphi]\bigm|\varphi\in E\}$. We need to see that $X{\models}\bigvee E$, so suppose otherwise. Then for all sufficiently large $\xi<\tau$, $$M_\xi{\models}\text{let }Y=V_\beta^{\widetilde{M}_\chi};\text{ then 
}{\mathrm{Col}}({\omega},V_\delta){\dststile{}{}}\exists X'{\subseteq}\check{Y}\text{ such that 
}X'{\models}\neg\bigvee E,$$ (as witnessed by $X$, by induction and choice of $\delta$), which gives that $$M_\alpha{\models}\text{``let 
}Y=V_{\beta^*}^{\widetilde{M}_{\chi^*}};\text{ then }{\mathbb Q}{\dststile{}{}}\exists 
X'{\subseteq}\check{Y}\text{ such that }X'{\models}(\neg\bigvee E)^*.$$ So $(\neg\bigvee E)\in{\mathcal{L}}'$, so $[\neg\bigvee E]\in{\mathbb P}$.

Work in $M_\tau$. By the density of $D$ and since $[\neg\bigvee E]\in{\mathbb P}$, there is $\varphi\in E$ such that $[\varphi]$ is compatible with $[\neg\bigvee E]$. Let $[\psi]\in{\mathbb P}$ with $[\psi]\leq[\varphi]$ and $[\psi]\leq[\neg\bigvee E]$. Let $G$ be ${\mathrm{Col}}({\omega},V_{\delta^*})$-generic. Then in $V[G]$, there is $X'$ such that $X'{\models}\psi^*$, and by the definition of $\leq$, therefore $X'{\models}\varphi^*$ and $X'{\models}(\bigvee E)^*$. But as before, the elementarity of $\widetilde{i}_{0\chi^*}$ then gives that $X'{\models}\neg\varphi^*$, a contradiction.

We can deduce that there are some interesting set-forcing extensions under ${\mathrm{ZFR}}$:

Suppose $(V,j){\models}{\mathrm{ZFR}}$. Let $\kappa={\mathrm{cr}}(j)$ and $\lambda=\kappa_{{\omega},j}$. Let $\alpha\in{\mathrm{OR}}$. Then there are partial orders ${\mathbb P}_i\in V$, for $i=0,\ldots,$ such that:

1.  ${\mathbb P}_0{\dststile{}{}}V_{\check{\kappa}}=\check{V_\kappa}$ and there is a rank-into-rank embeding $k:V_{\check{\kappa}}\to V_{\check{\kappa}}$ such that

    1.  $\kappa_{{\omega},k}=\check{\kappa}$ (hence $\operatorname{cof}(\check{\kappa})={\omega}$) and

    2.  $(\check{V_\lambda},\check{j}{\!\upharpoonright\!}\check{V_\lambda})$ is the ${\omega}$th iterate of $(\check{V_\kappa},k)$.

2.  for each $\beta<\kappa$ there is $\bar{\lambda}\in(\beta,\kappa)$ such that ${\mathbb P}_1$ forces that $V_{\check{\bar{\lambda}}}=\check{V_{\overline{\lambda}}}$ and $(\bar{\lambda}^+)=\check{\kappa}$ and there is a rank-into-rank embedding $k:V_{\check{\bar{\lambda}}}\to 
    V_{\check{\bar{\lambda}}}$ such that $(\check{V_\lambda},\check{j}{\!\upharpoonright\!}\check{V_\lambda})$ is the $\check{\kappa}$th iterate of $(V_{\check{\bar{\lambda}}},k)$.

The corresponding things are forceable over $M_{\omega}$, and over $M_{(\lambda^+)}$ with respect to $\lambda$, respectively. So the elementarity of $i_{0{\omega}}$ and $i_{0(\lambda^+)}$ pulls it back to $M_0=(V,j)$.

Suppose $(V,j){\models}{\mathrm{ZFR}}$ and let $\alpha$ be a limit ordinal and $\xi<\alpha$. Let $x\in\bigcap_{\xi<\alpha} M_\xi$. Then $x$ is *$(\xi,\alpha)$-stable* iff $i_{\xi\gamma}(x)=x$ for all $\gamma\in[\xi,\alpha)$.

Suppose $(V,j){\models}{\mathrm{ZFR}}$ and let $\alpha$ be a limit ordinal and $\eta={\mathrm{lot}}(\alpha)$. Let $J=\bigcap_{\xi<\alpha}N_\xi$. Let $\xi<\alpha$ be such that $\eta$ is $(\xi,\alpha)$-stable and either

1.  \[case:eta&lt;crit\] $\eta<{\mathrm{cr}}(j_\xi)$, or

2.  \[case:eta=sup\] ${\mathrm{cr}}(j_\xi)<\eta=\alpha=\sup_{\gamma<\alpha}{\mathrm{cr}}(j_\gamma)$.

Let $\mu=\operatorname{cof}^{M_\xi}(\eta)$, so $i_{\xi\gamma}(\mu)=\operatorname{cof}^{M_\gamma}(\eta)$ for all $\gamma\in[\xi,\alpha)$. Then:

1.  \[item:J\_sats\_ZF\] $J$ is transitive, proper class with $N_\alpha{\subseteq}J$ and $(J,M_\alpha){\models}{\mathsf{ZF}}$.

2.  \[item:mu&gt;om\] if $\mu>{\omega}$ then $N_\alpha=J$.

3.  \[item:mu=om\] if $\mu={\omega}$ then $N_\alpha{\subsetneq}J{\models}$“$\operatorname{cof}(\gamma)={\omega}$ for all $N_\alpha$-regular $\gamma\in[\kappa_\alpha,\kappa_{\alpha+{\omega}}]$”.

Part \[item:J\_sats\_ZF\]: $J$ is clearly a transitive with $N_\alpha{\subseteq}J$. And $(J,M_\alpha)$ is a class of $M_\xi=(N_\xi,j_\xi)$ for each $\xi<\alpha$. So $J$ satisfies Powerset: if $X\in J$ and $\xi<\alpha$ then because $J$ is a class of $M_\xi$, we have ${\mathcal{P}}(X){\cap}J\in M_\xi$; therefore ${\mathcal{P}}(X){\cap}J\in J$. Likewise, for each $\alpha\in{\mathrm{OR}}$, we have $V_\alpha{\cap}J\in J$ and $\left<V_\beta{\cap}J\right>_{\beta<\alpha}\in J$. Separation (with respect to the predicate $M_\alpha$) is similar to Powerset, and Collection follows from the preceding remarks.

Part \[item:mu&gt;om\]: Suppose $\mu>{\omega}$. Let $x\in J=\bigcap_{\gamma<\alpha}N_\gamma$. We have to see that $x$ is $\alpha$-es. Suppose not. Then for each $\gamma<\alpha$ there is $\gamma'\in(\gamma,\alpha)$ such that $i_{\gamma\gamma'}(x)\neq x$.

Work in $M_\xi$, where $\operatorname{cof}(\eta)=\mu>{\omega}$. Recall that $\xi+\eta=\alpha$. Define a sequence $\left<\xi_n\right>_{n<{\omega}}$ as follows: Set $\xi_0=0$. Given $\xi_n$, let $\xi_{n+1}$ be the least $\xi'\in(\xi_n,\eta)$ such that $\widetilde{i}^{M_\xi}_{\xi_n\xi'}(x)\neq x$. Then since $\operatorname{cof}(\eta)>{\omega}$, we have $(\sup_{n<{\omega}}\xi_n)=\xi_{\omega}<\eta$. So $x\in\widetilde{N}_{\xi_{\omega}}$, so by Lemma \[lem:star\_general\], $x$ is $\xi_{\omega}$-es, which contradicts the properties of $\left<\xi_n\right>_{n<{\omega}}$.

Part \[item:mu=om\]: Suppose $\mu={\omega}$. Work in $M_\xi$. Let $\left<\xi_n\right>_{1\leq n<{\omega}}$ be strictly increasing with $\sup_{n<{\omega}}=\eta$. Define a new sequence $\left<\xi'_n\right>_{0\leq n<{\omega}}$ recursively as follows. Set $\xi'_0=0$. Given $\xi'_n$, let $$\xi'_{n+1}=\widetilde{i}^{M_\xi}_{0\xi'_n}(\xi_{n+1}).$$ Clearly $\xi_n\leq\xi'_n<\xi'_{n+1}<\eta$, since $\eta$ is $(\xi,\alpha)$-stable. And $$\sup_{n<{\omega}}\xi'_n=\sup_{n<{\omega}}\xi_n=\eta.$$ Note that $\left<\xi'_n\right>_{n>m}\in M_{\xi'_m}$, so in fact $\left<\xi'_n\right>_{n<{\omega}}\in J$.

So we may assume $\left<\xi_n\right>_{n<{\omega}}\in J$, with $\xi_0=0$. Now let $\gamma\in[\kappa_\alpha,\kappa_{\alpha+{\omega}}]$ be regular in $N_\alpha$. We must see that $J{\models}$“$\operatorname{cof}(\gamma)={\omega}$”. We may assume that $\gamma\in{\mathrm{rg}}(i_{\xi\alpha})$; let $i_{\xi\alpha}(\bar{\gamma})=\gamma$. So $M_\xi{\models}$“$\bar{\gamma}$ is regular”. Now let $$\bar{\gamma}_n=i_{\xi,\xi+\xi_n}(\bar{\gamma}),$$ so $$i_{\xi+\xi_n,\xi+\xi_{n+1}}(\bar{\gamma}_n)=\bar{\gamma}_{n+1}$$ and $$i_{\xi+\xi_n,\alpha}(\bar{\gamma}_n)=\gamma.$$ Let $$\bar{\zeta}_{n+1}=\sup i_{\xi+\xi_n,\xi+\xi_{n+1}}``\bar{\gamma}_n.$$ Now $\bar{\zeta}_{n+1}<\bar{\gamma}_{n+1}$. For $j_{\xi+\xi_n}(\gamma')>\gamma'$, for all $\gamma'\in[\kappa_{\xi+\xi_n},\kappa_{\xi+\xi_n+{\omega}}]$, so in particular $$j_{\xi+\xi_n}(\bar{\gamma}_n)>\bar{\gamma}_n.$$ But $j_{\xi+\xi_n}:M_{\xi+\xi_n}\to M_{\xi+\xi_n}$, so by the regularity of $\bar{\gamma}_n$ in $M_{\xi+\xi_n}$, and since $j_{\xi+\xi_n}{\!\upharpoonright\!}{\mathrm{OR}}$ is amenable to $M_{\xi+\xi_n}$, we have $$(\sup j_{\xi+\xi_n}``\bar{\gamma}_n)<j_{\xi+\xi_n}(\bar{\gamma}_n).$$ But since $i_{\xi+\xi_n,\xi+\xi_{n+1}}=k{\circ}j_{\xi+\xi_n}$ for some embedding $k$, therefore $\bar{\zeta}_{n+1}<\bar{\gamma}_{n+1}$.

Now let $\zeta_{n+1}=i_{\xi+\xi_{n+1},\alpha}(\bar{\zeta}_{n+1})$. Then easily $\sup_{n<{\omega}}\zeta_{n+1}=\gamma$, and note that $\left<\zeta_{n+1}\right>_{n<{\omega}}\in J$, since $\left<\xi_n\right>_{n<{\omega}}\in J$.

\[tm:N\_tau\_not\_set-ground\] Suppose $(V,j){\models}{\mathrm{ZFR}}$. Let $\tau\geq{\omega}$. Then:

1.  \[item:j\_rest\] For $\alpha\in{\mathrm{OR}}$, we have $i_{0{\omega}}{\!\upharpoonright\!}\alpha\in 
    N_\tau$ iff $\alpha<\kappa_{\omega}$.

2.  \[item:not\_proper\_ground\] $N_\tau$ is not a set-ground of $V$.

Part \[item:j\_rest\]: Consider first $\tau={\omega}$.

Let $\alpha<\kappa_{\omega}$. Then $i_{0n}{\!\upharpoonright\!}\alpha\in N_n=V$ for all $n<{\omega}$. But then $$i_{0{\omega}}{\!\upharpoonright\!}\alpha=i_{n{\omega}}(i_{0n}{\!\upharpoonright\!}\alpha)\in N_{\omega},$$ as desired. Conversely, $i_{0{\omega}}{\!\upharpoonright\!}\kappa_{\omega}\notin N_{\omega}$ since $i_{0{\omega}}``\kappa_{\omega}$ is cofinal in $\kappa_{{\omega}+{\omega}}$, and $\operatorname{cof}^{N_{\omega}}(\kappa_{\omega})=\kappa_{\omega}$, because $\kappa_{\omega}={\mathrm{cr}}(j_{\omega})$, while $\operatorname{cof}^{N_{\omega}}(\kappa_{{\omega}+{\omega}})={\omega}$, because $\kappa_{{\omega}+{\omega}}=\widetilde{\kappa}_{\omega}^{M_{\omega}}$.

Since ${\mathcal{P}}({<\kappa_{{\omega}+{\omega}}}){\cap}N_\tau={\mathcal{P}}({<\kappa_{{\omega}+{\omega}}}){\cap}N_{\omega}$ and ${\mathcal{P}}(\kappa_{{\omega}+{\omega}}){\cap}N_\tau{\subseteq}{\mathcal{P}}(\kappa_{{\omega}+{\omega}})$ for all $\tau>{\omega}$, this establishes part \[item:j\_rest\].

Part \[item:not\_proper\_ground\]:

$N_{\omega}$ is not a set ground of $V$.

Suppose not. It suffices to see that $i_{0{\omega}}{\!\upharpoonright\!}\alpha\in N_{\omega}$ for each $\alpha\in{\mathrm{OR}}$, as then with $\alpha\geq\kappa_\kappa$, this contradicts part \[item:j\_rest\]. Let ${\mathbb P}\in N_{\omega}$ be such that $N_{\omega}$ is a ground of $V$ via $G{\subseteq}{\mathbb P}$. Let $\alpha\in{\mathrm{OR}}$ with ${\mathbb P}\in V_\alpha$. Let $\beta\in{\mathrm{OR}}$ be least such that $M_{\omega}$ has no surjection $(V_\alpha)^{<{\omega}}\to\beta$. Let $\sigma\in 
M_{\omega}$ be such that $\sigma_G=i_{0{\omega}}{\!\upharpoonright\!}\beta$. Then there is $X{\subseteq}\beta$ of ordertype $\geq\alpha$ and $p\in G$ such that $p$ decides $\sigma{\!\upharpoonright\!}\check{X}$. Thus, $k=i_{0{\omega}}{\!\upharpoonright\!}X\in M_{\omega}$ and $p{\dststile{}{}}\sigma{\!\upharpoonright\!}\check{X}=\check{k}$. Let $\gamma={\mathrm{ot}}(X)$ and $\pi:\gamma\to X$ the increasing enumeration. So $\pi\in M_{\omega}$. Applying $i=i_{0{\omega}}$, we get $i(\pi):i(\gamma)\to i(X)$ is the increasing enumeration, and $i(\pi),i(\gamma),i(X)\in M_{\omega}$. But $i{\circ}\pi=i(\pi){\circ}i{\!\upharpoonright\!}\gamma$, so $i{\!\upharpoonright\!}\gamma\in M_{\omega}$. Since $\alpha\leq\gamma$, we have $i{\!\upharpoonright\!}\alpha\in M_{\omega}$, as desired.

But then if $\tau>{\omega}$ and $N_\tau$ is a set-ground of $V$, then letting $X\in V$ be such that $N_\tau[X]=V$, then since $X$ is set-generic over $N_{\omega}$ by Theorem \[tm:every\_set\_generic\_over\_N\_tau\], $N_{\omega}[X]$ is a set-generic extension of $N_{\omega}$, but since $N_\tau{\subseteq}N_{\omega}$, then $N_{\omega}[X]=V$, so $N_{\omega}$ is a set-ground of $V$, contradicting the claim.

In the following, we define ${\mathrm{HOD}}(X)={\mathrm{Hull}}^V({\mathrm{OR}}\cup X)$; that is, the collection of all sets which are definable over $V$ from (finitely many) ordinals and parameters in $X$.

It is shown in [@gen_kunen_incon Theorem 12] that if $V{\models}{\mathsf{ZFC}}+$“$j:V\to M$ is an elementary embedding” (with $M$ wellfounded) then $V={\mathrm{Hull}}^V({\mathrm{rg}}(j))$. But the proof relies strongly on ${\mathsf{AC}}$. It trivially fails with respect to $j$ when $(V,j){\models}{\mathrm{ZFR}}$ (we have ${\mathrm{rg}}(j)={\mathrm{Hull}}^V({\mathrm{rg}}(j))$ by elementarity). It also fails with respect to $j=i_{0{\omega}}$ (but since $N_{\omega}\neq V$, this time it is not trivial then the conclusion fails, as shown in the following theorem).

\[tm:V=HOD(X)\_no\_R\] Suppose $(V,j){\models}{\mathrm{ZFR}}$. Then there is no set $X\in V$ such that $V={\mathrm{HOD}}(X)$. In fact there is no set $X$ such that $V={\mathrm{Hull}}^V(N_{\omega}\cup X)$. Therefore $V\neq{\mathrm{Hull}}^V({\mathrm{rg}}(i_{0{\omega}}))$.

Suppose $X\in V$ is such that $V={\mathrm{Hull}}^V(N_{\omega}\cup X)$. Let $k_n=i_{n{\omega}}{\!\upharpoonright\!}X^{<{\omega}}$, for $n<{\omega}$. So ${\mathrm{rg}}(k_n){\subseteq}N_{\omega}$. Let $\vec{k}=\left<k_n\right>_{n<{\omega}}$. Then $(X,\vec{k})\in V$, so by Theorem \[tm:every\_om\], we have $(X,\vec{k})\in 
N_{\omega}[G]$ for some set-generic extension $N_{\omega}[G]$ of $N_{\omega}$. We will show that $N_{\omega}[G]=V$, so $N_{\omega}$ is a set-ground of $V$, contradicting Theorem \[tm:N\_tau\_not\_set-ground\].

Let $Y\in V$. Then by our contradictory assumption, there is $\alpha\in{\mathrm{OR}}$ and $n<{\omega}$ such that $Y$ is encoded into $t=\operatorname{Th}_{\Sigma_n}^V((N_{\omega}{\cap}V_\alpha)\cup X)$. So it suffices to see that $t\in N_{\omega}[G]$. We certainly have $(N_{\omega}{\cap}V_\alpha)\cup X\in N_{\omega}[G]$, so we just need to verify the appropriate instance of Separation.

So let $\varphi$ be a $\Sigma_n$ formula and $\vec{y}\in(N_{\omega}{\cap}V_\alpha)^{<{\omega}}$ and $\vec{x}\in X^{<{\omega}}$. Then for each $m<{\omega}$, we have $$[V=N_m{\models}\varphi(\vec{y},\vec{x})]\iff 
[N_{\omega}{\models}\varphi(i_{m{\omega}}(\vec{y}),i_{m{\omega}}(\vec{x}))].$$ But $i_{m{\omega}}(\vec{x})=k_m(\vec{x})$ for all $m<{\omega}$, and $i_{m{\omega}}(\vec{y})=\vec{y}^*=i_{{\omega},{\omega}+{\omega}}(\vec{y})$ for all sufficiently large $m<{\omega}$. Therefore $$[V{\models}\varphi(\vec{y},\vec{x})]\iff
[N_{\omega}{\models}\exists \ell<{\omega}{\forall}m\geq\ell\ 
[\varphi(\vec{y}^*,k_m(\vec{x}))]].$$ Since $\left<k_m\right>_{m<{\omega}}\in N_{\omega}[G]$ and $i_{{\omega},{\omega}+{\omega}}{\!\upharpoonright\!}(N_{\omega}{\cap}V_\alpha)\in N_{\omega}[G]$, it easily follows that $t\in N_{\omega}[G]$, as desired.[^7]

Suppose $(V,j){\models}{\mathrm{ZFR}}$. Can it be that $V={\mathrm{Hull}}^V({\mathrm{OR}}\cup{\mathrm{rg}}(j))$? Or $V={\mathrm{Hull}}^V(M_{\omega}\cup{\mathrm{rg}}(j))$? Etc. Obviously $V\neq{\mathrm{Hull}}^V({\mathrm{rg}}(j))$. Can ${\mathsf{AC}}$ be set-forceable over $V$? (Note that if $V={\mathrm{HOD}}(X)$ for a set $X$, then ${\mathsf{AC}}$ is forced by ${\mathrm{Col}}({\omega},X)$.)

Assume ${\mathsf{ZF}}$. We recall the definition of the *Vopenka algebra* ${\mathrm{Vop}}$ for adding reals to ${\mathrm{HOD}}$. First define ${\mathrm{Vop}}'$ in $V$ as follows. Conditions are non-empty ${\mathrm{OD}}$ subsets of ${\mathbb R}$; the ordering is $A\leq B\iff A{\subseteq}B$. Then ${\mathrm{Vop}}\in{\mathrm{HOD}}$ is the natural coding of ${\mathrm{Vop}}'$ as a set of ordinals. Let $\pi:{\mathrm{Vop}}\to{\mathrm{Vop}}'$ be the natural isomorphism.

Now given $x\in{\mathbb R}$, we define the filter $G_x=\{p\in{\mathrm{Vop}}\bigm|x\in\pi(p)\}$.

\[fact:Vopenka\] $G_x$ is $({\mathrm{HOD}},{\mathrm{Vop}})$-generic and $x\in{\mathrm{HOD}}[G_x]$; in fact, ${\mathrm{HOD}}[G_x]={\mathrm{HOD}}_x$.

We give the standard proofs. It is clear that $G_x$ is a filter. For genericity, let $D{\subseteq}{\mathrm{Vop}}$ be dense, with $D\in{\mathrm{HOD}}$. Let $D'=\pi``D$. Then $D'\in{\mathrm{OD}}$, so $\bigcup D'\in{\mathrm{Vop}}'$. We have $\bigcup D'={\mathbb R}$, since otherwise $A={\mathbb R}{\backslash}\bigcup D'\in{\mathrm{Vop}}'$ and clearly $\pi^{-1}(A)$ is incompatible with every element of $D$, a contradiction. Therefore $x\in\bigcup D'$, so $x\in A$ for some $A\in D'$, so $x\in\pi(p)$ for some $p\in D$, so $G_x{\cap}D\neq\emptyset$.

Clearly ${\mathrm{HOD}}[G_x]{\subseteq}{\mathrm{HOD}}_x$. Now let $\varphi$ be a formula and $\eta,\gamma\in{\mathrm{OR}}$ and $$X=\{\xi<\gamma\bigm|\varphi(x,\xi,\eta)\}.$$ We must see that $X\in{\mathrm{HOD}}[G_x]$. For $\xi<\gamma$ let $$t'_\xi=\{y\bigm|y\in{\mathbb R}\text{ and 
}\varphi(y,\xi,\eta)\}.$$ So $t'_\xi\in{\mathrm{Vop}}'$. Let $\pi(t_\xi)=t'_\xi$. Note that $\left<t_\xi\right>_{\xi<\gamma}\in{\mathrm{HOD}}$. It is straightforward to show that $$t_\xi\in G_x\iff\varphi(x,\xi,\eta)\iff\xi\in X,$$ so $X\in{\mathrm{HOD}}[G_x]$, as desired.

Although ${\mathrm{ZFR}}$ implies that ${\mathrm{HOD}}{\subseteq}M_{\omega}$, we have:

Suppose $(V,j){\models}{\mathrm{ZFR}}$. Then ${\mathrm{HOD}}\not{\subseteq}\bigcap_{\alpha\in{\mathrm{OR}}}M_\alpha$.

Let $R=(V_{\kappa_{\omega}},j{\!\upharpoonright\!}V_{\kappa_{\omega}})$. Note that $R$ is iterable and for each $\alpha$, writing $R_\alpha$ for the $\alpha$th iterate and $\lambda=\widetilde{\kappa_{\omega}}^{M_\alpha}$, we have $$R_\alpha=(V_\lambda^{M_\alpha},j_\alpha{\!\upharpoonright\!}V_{\lambda}^{M_\alpha}).$$

Let $G$ be $(V,{\mathrm{Col}}({\omega},V_{\kappa_{\omega}}))$-generic. Let $H={\mathrm{HOD}}^{V[G]}$. By the homogeneity of the collapse, $H{\subseteq}{\mathrm{HOD}}$. Let $x\in V[G]$ be a real coding $R$. Then by Fact \[fact:Vopenka\], $R\in H[G_x]=({\mathrm{HOD}}^{V[G]}_x)$ is a set-generic extension of $H$.

Let $\xi\in{\mathrm{OR}}$ be such that ${\mathrm{Vop}}^{V[G]}{\subseteq}\xi$. So $(\xi^{+n+1})^{H[G_x]}=(\xi^{+n+1})^H$. Since $R\in{\mathrm{HC}}^{H[G_x]}$, in $H[G_x]$ we can iterate $R$, and $R_{(\xi^+)^{H}}$ has cardinality $(\xi^+)^{H}$ there, with $(\xi^+)^H={\mathrm{cr}}(j_{(\xi^+)^H})$. So ${\mathcal{P}}((\xi^+)^H){\cap}R_\infty$ also has cardinality $(\xi^+)^H$ there. So $H\not{\subseteq}R_\infty$.

So $H{\subseteq}{\mathrm{HOD}}\not{\subseteq}\bigcap_{\alpha\in{\mathrm{OR}}}M_\alpha=\bigcap_{\alpha\in{\mathrm{OR}}}
R_\alpha$.

Suppose $(V,j){\models}{\mathrm{ZFR}}$. Then $${\mathrm{rg}}(i_{0{\omega}})=\{x\bigm|j(x)=x\text{ and }x{\subseteq}M_{\omega}\}$$ $$=\{x\bigm|j(x)=x\text{ and }x\in M_{\omega}\}.$$ Proof: We have $$j(i_{0{\omega}}(y))=i_{1{\omega}}(j(y))=i_{1{\omega}}{\circ}i_{01}(y)=i_{0{\omega}}(y).$$ Conversely, let $x\in M_{\omega}$ be such that $j(x)=x$. Since there is $n<{\omega}$ such that $x\in{\mathrm{rg}}(i_{n{\omega}})$, so $M_n{\models}$“$x\in{\mathrm{rg}}(\widetilde{i}_{0{\omega}})$” but $j(x)=x$, a fact which is propagated by $j$ and its iterates, i.e. $j_n(x)=x$ for all $n<{\omega}$, so $x=i_{0n}(x)$, so the fact above pulls back to $M_0$, so $x\in{\mathrm{rg}}(i_{0{\omega}})$.

Now suppose that $x{\subseteq}M_{\omega}$ and $j(x)=x$; we need to see that $x\in M_{\omega}$. But for $y\in M_{\omega}$, we have $$y\in x\iff y^*\in i_{0{\omega}}(x),$$ but $y^*=\widetilde{i}^{M_{\omega}}_{0{\omega}}(y)$ for $y\in M_{\omega}$, and $i_{0{\omega}}(x)\in M_{\omega}$, so we get $x\in M_{\omega}$.

Note then that ${\mathrm{rg}}(i_{n{\omega}})=\{\alpha\in{\mathrm{OR}}\bigm|j_n(\alpha)=\alpha\}$ for each $n<{\omega}$.

\[cor:V=HOD\_A(X)\_no\_tR\] Suppose there is a total Reinhardt cardinal. Then there is no class $A$ such that $V={\mathrm{HOD}}_A(X)$.

Just let $j:(V,(A,X))\to (V,(A,X))$ be elementary, and run the obvious variant of the Reinhardt proof.

No Berkeley cardinal in ${\mathrm{HOD}}_A(X)$ {#sec:Berkeley}
=============================================

For the definition of *Berkeley cardinal*, one should see [@woodin_koellner_bagaria].

\[tm:V=HOD\_A(X)\_no\_B\] Let $\Omega\in{\mathrm{OR}}$ be such that $V_\Omega{\models}{\mathsf{ZF}}+$“There is a Berkeley cardinal”. Then there is no $A{\subseteq}V_\Omega$ such that $(V_\Omega,A){\models}{\mathsf{ZF}}+$“$V={\mathrm{HOD}}_A(X)$ for some set $X$”.

Suppose otherwise and work in $(V_\Omega,A)$. Let $\beta$ be a Berkeley cardinal. Clearly $\beta$ is a limit of inaccessibles, so let $\kappa_0$ be inaccessible. Let $\lambda\in{\mathrm{OR}}$, $\lambda>\beta$, be such that $(V_\lambda,A{\cap}V_\lambda){\preccurlyeq}_n(V,A)$ and $\operatorname{cof}(\lambda)>{\omega}$ where $n=10$ or so (we only need $\kappa_0$ to see that there was some cofinality $>{\omega}$). Let $$j:(V_\lambda,A{\cap}V_\lambda)\to(V_\lambda,A{\cap}V_\lambda)$$ be elementary with ${\mathrm{cr}}(j)<\beta$. By elementarity, there is $X\in V_\lambda$ such that $(V_\lambda,A{\cap}V_\lambda){\models}$“$V={\mathrm{HOD}}_A(X)$”.

Now we adapt the proof in the Reinhardt cardinal context to $(V_\lambda,A{\cap}V_\lambda,j)$. We have that $j$ is amenable to $V_\lambda$; i.e. $j{\!\upharpoonright\!}V_\alpha\in V_\lambda$ for each $\alpha<\lambda$. We have that $j``\lambda$ is cofinal in $\lambda$ and $j``V_\lambda$ is $\in$-cofinal in $V_\lambda$. Consider the structure $M_0=(V_\lambda,A{\cap}V_\lambda,j_0)$ where $j=j_0$. Note that we can define $M_1=j_0(M_0)$ in the usual manner, i.e. $M_1=(V_\lambda,A{\cap}V_\lambda,j_1)$, where $j_1=j_0(j_0)=\bigcup_{\alpha<\lambda}j(j{\!\upharpoonright\!}V_\alpha)$. Also note that $V_\lambda=j_0(V_\lambda)=\bigcup_{\alpha<\lambda}j_0(V_\alpha)$ and likewise for $A{\cap}V_\lambda$. Note that there are unboundedly many $\alpha<\lambda$ of cofinality ${\omega}$ which are fixed by $j$. Fix such an $\alpha$. Note that $k=j_0{\!\upharpoonright\!}V_\alpha$ is fully elementary as a map $$k:(V_\alpha,A{\cap}V_\alpha,j_0{\!\upharpoonright\!}V_\alpha)\to(V_\alpha,A{\cap}V_\alpha,j_0(j_0{\!\upharpoonright\!}V_\alpha)).$$ Therefore $j:M_0\to M_1$ is cofinal $\Sigma_1$-elementary.

Also since these bounded fragments of $j_0$ are fully elementary, these facts are lifted by $j_0$, and so $$j_1:M_1\to M_2$$ is also cofinal $\Sigma_1$-elementary, etc. (In particular, note that $j_1$ also coheres $A$.) Thus, we can iterate this through the $M_n$’s.

Let $i_{mn}:M_m\to M_n$ be the composition $j_{n-1}{\circ}\ldots{\circ}j_m$. Given $\alpha<\lambda$, let $\alpha'=\sup_{n<{\omega}}i_{0n}(\alpha)$. Then $\alpha'\leq\lambda$, and since $\operatorname{cof}(\lambda)>{\omega}$, therefore $\alpha'<\lambda$. Therefore the sequence $\left<i_{0n}(j{\!\upharpoonright\!}V_\alpha)\right>_{n<{\omega}}{\subseteq}V_\lambda$, and so $\in V_\lambda$.

The map $\alpha\mapsto$ this sequence is $\Sigma_1$-definable over $M_0$. Therefore the usual proof shows that $M_{\omega}$ and the maps $i_{n{\omega}}:M_n\to M_{\omega}$ are uniformly $\Sigma_1$-definable over $M_n$, and $M_{\omega}$ is wellfounded, with ${\mathrm{OR}}^{M_{\omega}}=\lambda$.

We can now run the rest of the Reinhardt proof for a contradiction (of course using ${\mathrm{HOD}}_A(X)$ in place of ${\mathrm{HOD}}(X)$, which works because each $j_n$ coheres $A$).

Is it consistent that there is a Berkeley cardinal and ${\mathsf{AC}}$ can be forced?

Forcing extensions from ${\mathrm{ZFR}}$
========================================

\[rem:forcing\_ext\_from\_R\] Let $(M,j){\models}{\mathrm{ZFR}}$ and $\kappa={\mathrm{cr}}(j)$. Then for all $\alpha\in{\mathrm{OR}}^M$ there is $\eta\geq\alpha$ and a set-forcing ${\mathbb P}\in M$ which forces that in $M^{\mathbb P}$, $V_\kappa^{M^{\mathbb P}}=V_\kappa^M$, $M^{\mathbb P}{\models}$“There is an elementary $k:V_\eta\to V_\eta$ with $\kappa_{{\omega},k}=\kappa$”. This follows immediately from the fact that there are cofinally many $\eta\in{\mathrm{OR}}^M$ such that $j(\eta)=\eta$, and taking any such $\eta$, that there is $G\in M$ which is set-generic over $M_{\omega}$ and with $(V_\eta^M,j{\!\upharpoonright\!}V_\eta^M)\in M_{\omega}[G]$, and using the elementarity of $i_{0{\omega}}$ and that $i_{0{\omega}}(\kappa)=\lambda=\kappa_{{\omega},j}$, and noting that $V_\eta^{M_{\omega}[G]}=V_\eta^M$.

Also, for all $\alpha\in{\mathrm{OR}}$ there is $\eta\geq\alpha$ and $\bar{\lambda}<\kappa$ with $M{\models}$“$\bar{\lambda}$ is inaccessible”, and a set-forcing ${\mathbb P}\in M$ which forces that in $M^{\mathbb P}$, we have $V_{\bar{\lambda}}^{M^{\mathbb P}}=V_{\bar{\lambda}}^M$, $M^{\mathbb P}{\models}$“There is an elementary $k:V_\eta\to V_\eta$ with $\kappa_{\omega}(k)={\bar{\lambda}}$” and $(\bar{\lambda}^+)^{M^{\mathbb P}}=\kappa$. This is as above, but using $M_{(\lambda^+)^M}$ instead of $M_{\omega}$.

Assume $(V,j){\models}{\mathrm{ZFR}}+{\mathsf{DC}}_\lambda$ where $\lambda=\lambda_j$. Then it is straightforward to construct a forcing extension having an inner model satisfying ${\mathsf{ZFC}}+I_0+$“$V=L[A]\text{ for some set 
}A{\subseteq}{\mathrm{OR}}$”, in fact as witnessed by an embedding $$k:L(V_{\lambda+1})\to L(V_{\lambda+1})$$ (with $V_{\lambda+1}$ preserved) with $k{\!\upharpoonright\!}V_{\lambda+1}=j{\!\upharpoonright\!}V_{\lambda+1}$. For note first that (i) $V_\lambda$ is wellordered, in ordertype $\lambda$. (This is a standard argument: consider $\mu<\lambda$ and injections from $\alpha\leq\lambda$ into $V_\mu$; by ${\mathsf{DC}}_\lambda$, either there is an injection from $\lambda\to V_\mu$, which contradicts ${\mathrm{ZFR}}$, or a bijection between some $\alpha<\lambda$ and $V_\mu$, so we have the latter. Then with ${\mathsf{DC}}$, we can put an ${\omega}$-sequence of these together to make a bijection between $V_\lambda$ and $\lambda$.) Further, (ii) $\lambda^+$ is regular; (iii) we can force a wellorder $<^*$ of $V_{\lambda+1}$ in ordertype $\lambda^+$ without changing ${\mathcal{P}}(\lambda)$. So in $V[{<^*}]$, we have the same $L(V_{\lambda+1})$, and $j$ restricts to give an elementary $$k=j{\!\upharpoonright\!}L(V_{\lambda+1}):L(V_{\lambda+1})\to 
L(V_{\lambda+1}).$$ Now we can wellorder the elements of $L(V_{\lambda+1})$ as usual from $<^*$ (every element of $L(V_{\lambda+1})$ is definable from ordinals and elements of $V_{\lambda+1}$). Therefore $k$ is coded by a class $C$ of ordinals of $V[{<^*}]$. Finally, take $X=V_\eta^{L(V_{\lambda+1})}$ with $\eta$ large enough, and $k'=k{\!\upharpoonright\!}X$. Then $L(V_{\lambda+1},{<^*},k')$ is of the form $L[A]$ with a set of ordinals $A$, and models ${\mathsf{ZFC}}+I_0$. because letting $E=E_{k{\!\upharpoonright\!}X}$ be the extender derived from $k'$, we can form ${\mathrm{Ult}}(L(V_{\lambda+1}),E_{k{\!\upharpoonright\!}X})$, and use the methods of §\[sec:no\_Reinhardt\_in\_L(X)\] to see this works.

Applying the foregoing to iterates $M_\alpha$ of $(V,j)$, and then arguing further as in Remark \[rem:forcing\_ext\_from\_R\], we can conclude that there are set-forcing extensions which have inner models $M$ satisfying ${\mathsf{ZFC}}+I_0$, as witnessed by an embedding $$k:L(V_{\bar{\lambda}+1})\to 
L(V_{\bar{\lambda}+1})$$ with either $\bar{\lambda}=\kappa={\mathrm{cr}}(j)$, or $\bar{\lambda}<\kappa$, as desired, and such that $V_{\bar{\lambda}}^M=
V_{\bar{\lambda}}$.

Reinhardts and small forcing {#sec:small_forcing}
============================

For a set $X$, *$X$-${\mathsf{AC}}$* is the statement that for every function $f$ with domain $X$, there is a choice function for $f$.

The following is a straightforward adaptation of the standard ground definability arguments of Laver, Woodin and Hamkins; it is probably similar to the adaptation of Gitman and Johnstone in [@ground_def_gitman_johnstone].

\[tm:enough\_AC\_ground\_def\] Let $M{\models}{\mathsf{ZF}}$. Let $M{\subseteq}M[G]$ be a set generic extension via ${\mathbb P}{\subseteq}B=V_\eta^M$. Suppose that $B$ is wellorderable in $M[G]$ and let $\gamma=\operatorname{card}^{M[G]}(B)$. Suppose $M[G]{\models}$“$\gamma^+$-${\mathsf{DC}}$”.

Then $M$ is definable over $M[G]$ from the parameter $V_{\eta+1}^M$.

Work in $M[G]$. Let $\lambda>\eta$ be an ordinal which is a strong limit cardinal, in that there is no surjection $f:V_\alpha\to\lambda$ for any $\alpha<\lambda$.

The following claim completes the proof:

$V_\lambda{\cap}M$ is the unique transitive model $N{\subseteq}V_\lambda$ such that

1.  $\lambda={\mathrm{OR}}{\cap}N$,

2.  $V_\alpha{\cap}N\in N$ for each $\alpha<\lambda$,

3.  $N{\models}$ Separation, Pairing, Union, Powerset,

4.  $V_{\eta+1}^N=V_{\eta+1}^M$,

5.  There is $({\mathbb Q},H)$ such that ${\mathbb Q}\in V_{\eta+1}^N=V_{\eta+1}^M$ and $H$ is $(N,{\mathbb Q})$-generic and $V_\lambda=N[H]$.

We show by induction on $\xi>\eta$ that $V_\xi^N=V_\xi^M$. This is by assumption for $\xi=\eta+1$, and trivial by induction if $\xi$ is a limit. So assume it holds at $\xi<\lambda$.

Given $A{\subseteq}N$ and $B\in N$, we say that $A$ is *$B$-amenable to $N$* iff for every function $\pi\in N$ with ${\mathrm{dom}}(\pi)=B$, we have $A{\cap}{\mathrm{rg}}(\pi)\in N$.

Let $A{\subseteq}V_{\xi}$. We must show that $A\in M\iff A\in N$. We do this by proving the following, with $B=V_\eta^M=V_\eta^N$:

1.  \[item:A\_in\_M\] $A\in M$ iff $A$ is $B$-amenable to $M$,

2.  \[item:A\_in\_N\] $A\in N$ iff $A$ is $B$-amenable to $N$,

3.  \[item:pi\_in\_M\_N\] for all $\pi:B\to V_\xi^M=V_\xi^N$, we have $\pi\in M$ iff $\pi\in N$.

Putting these things together, we get $A\in M$ iff $A\in N$, as desired.

Note first that we have the Forcing Theorem with respect to $\Sigma_0$ formulas, and definability of the Forcing Relation for $\Sigma_0$ formulas, over $V_\lambda^M$ and $V_\lambda^N$; this follows from Separation and that $\lambda$ is large enough.

Parts \[item:A\_in\_M\],\[item:A\_in\_N\]: Cearly it suffices to consider part \[item:A\_in\_N\]. For the non-trivial direction, suppose that $A\notin N$; we will show that $A$ is not $B$-amenable to $N$. We have $V_\lambda=N[H]$. So let $\dot{A}\in N$ be such that $\dot{A}_H=A$. Since $A{\subseteq}V_\xi^N$ and $A\notin N$, there is $p_0\in H$ forcing “$\dot{A}{\subseteq}\check{V_\xi}\text{ but }\dot{A}\notin\check{V}$”.

Now in $V$, where ${\mathbb Q}$ is wellordered and we have $|{\mathbb Q}|^+$-${\mathsf{AC}}$, let $f:{\mathbb Q}\to V_\xi^N$ be such that for each $p\leq p_0$, $f(p)$ is some $x$ such that $$N{\models}p\not{\dststile{}{}}\check{x}\in\dot{A}\text{ and 
}p\not{\dststile{}{}}\check{x}\notin\dot{A}.$$

Let $\dot{f}\in N$ be such that $\dot{f}_H=f$. Let $p_1\in H$ be such that in $N$, $p_1$ forces “$\dot{f}:\check{{\mathbb Q}}\to\check{V_\xi}$, and for each $p\leq\check{p_0}$, $\check{V}$ thinks that $p$ does not decide whether $\dot{f}(p)\in\dot{A}$”.

Working again in $N$, define the partial function $f_*$ with domain ${\subseteq}{\mathbb Q}{\times}{\mathbb Q}$, where for $(p,q)\in{\mathbb Q}{\times}{\mathbb Q}$, if $q\leq p\leq p_1$ then $f_*(p,q)$ is the unique $x$ such that $q{\dststile{}{}}\dot{f}(\check{p})=\check{x}$, if there is such an $x$, and $f_*(p,q)$ is undefined otherwise.

Let $C={\mathrm{rg}}(f_*)\in N$. We claim that $C{\cap}A\notin N$, which completes the proof of part \[item:A\_in\_N\]. So let $D{\subseteq}C$ with $D\in N$. For each $p\leq p_1$ there is $q\leq p$ and $x$ such that $f_*(p,q)$ is defined and $f_*(p,q)=x$, and so $x\in C$; but then there are also $p'\leq p$ and $p''\leq p$ such that $p'{\dststile{}{}}\check{x}\in\dot{A}$ and $p''{\dststile{}{}}\check{x}\notin\dot{A}$. So by density, there will be some such $x$ such that $x\in A\iff x\notin D$. So $C{\cap}A\neq D$, as desired.

Part \[item:pi\_in\_M\_N\]:

Let $\pi^N:B\to V_\xi^M=V_\xi^N$, with $\pi^N\in N$. Let $\pi^M:B\to V_\xi^M$, with $\pi^M\in M$. We show that $\pi^N\in M$ and $\pi^M\in N$.

Work in $M[G]$, where $|B|=\gamma$ and $\gamma^+$-${\mathsf{DC}}$ holds. Construct a continuous increasing sequence $\left<X_\alpha\right>_{\alpha<\gamma^+}$ of sets with the following properties:

1.  $X_0={\mathrm{rg}}(\pi^N)$

2.  $X_0\cup{\mathrm{rg}}(\pi^M){\subseteq}X_1$,

3.  $X_{{\omega}\alpha+2n+1}\in M$

4.  $X_{{\omega}\alpha+2n+2}\in N$

5.  $X_\alpha{\subseteq}V_\xi^M=V_\xi^N$,

6.  $|X_\alpha|\leq\gamma$

7.  for each limit $\alpha$, $X_\alpha$ is extensional.

Given $X=X_{{\omega}\alpha+2n}$, first let $Y{\subseteq}V_{\xi}^M$ be some set with $X{\subseteq}Y$ and $|Y|=\gamma$ and such that for all $x,y\in X$, if $x\neq y$ then there is $z\in Y$ such that $z\in x$ iff $z\notin y$. Then let $X_{{\omega}\alpha+2n+1}\in M$ be some $Y'$ such that $Y{\subseteq}Y'{\subseteq}V_\xi^M$ and $Y'$ is the surjective image of $B$ in $M$; this exists because $V_\lambda=V_\lambda^M[G]$, by which $M$ has a name $\dot{Y}$ for $Y$, and so $M$ can find a small covering set. For $X_{{\omega}\alpha+2n+2}$ it is likewise. The sequence is continuous. By $\gamma^+$-${\mathsf{DC}}$, we can proceed through $\gamma^+$ sets; note that $\gamma$-${\mathsf{DC}}$ ensures here that at limit stages $\alpha$, we still have $|X_\alpha|=\gamma$.

Now since $\lambda$ is large, the sequence is in $V_\lambda$. So let $\dot{X}^N\in N$ and $\dot{X}^M\in V_\lambda^M$ be names for it. Then we can find $p\in G$ such that for cofinally many $\alpha<\gamma^+$, in $M$, $p$ decides the value of $\dot{X}_\alpha$, and forces that the sequence is continuous. Let $D^M\in M$ be this cofinal set. Then $\left<X_\alpha\right>_{\alpha\in D}\in M$. Note that $D^M$ is closed. We get a similar club $D^N\in N$. Let $\alpha\in D^M{\cap}D^N$ be a limit ordinal. Then $X_\alpha\in M{\cap}N$, $X_\alpha$ is extensional, and $M,N$ both have surjections $B\to X_\alpha$ (since they have surjections to $X_{\alpha+1}$ and $X_{\alpha+2}$ respectively).

Since $X_\alpha$ is extensional, we can let $\bar{X}$ be the transitive collapse of $X_\alpha$, and $\pi:\bar{X}\to X_\alpha$ the uncollapse map. Then $\bar{X},\pi\in M{\cap}N$. Note that $N,M$ have surjections from $B$ to $\bar{X}$. Since $\bar{X}$ is transitive, it follows that $\bar{X}$ is coded by subsets of $B$ in $M,N$. But ${\mathcal{P}}(B){\cap}M={\mathcal{P}}(B){\cap}N$. Let $\sigma:B\to\bar{X}$ be a common surjection. Because we have $\sigma$, we get ${\mathcal{P}}(\bar{X}){\cap}M={\mathcal{P}}(\bar{X}){\cap}N$.

We now show that $\pi^M\in N$. We have $\pi^M:B\to V_\xi^M$ and ${\mathrm{rg}}(\pi^M){\subseteq}{\mathrm{rg}}(\pi)$. Let $\bar{\pi}^M:B\to\bar{X}$ be $\bar{\pi}^M=\pi^{-1}{\circ}\pi^M$. Then easily $\bar{\pi}^M\in N$, so $\pi^M=\pi{\circ}\bar{\pi}^M\in N$.

The fact that $\pi^N\in M$ is proved in the same way. This completes the proof of the claim.

The claim completes the proof of the theorem.

\[tm:Reinhardt\_with\_enough\_AC\_restricts\] Let $V{\subseteq}V[G]$ be a set forcing extension via forcing ${\mathbb P}$, $(V[G],j){\models}{\mathrm{ZFR}}$ with ${\mathbb P}{\subseteq}V_{\eta}$ where $\eta<\kappa={\mathrm{cr}}(j)$, and suppose that $V[G]{\models}$“$X$-${\mathsf{AC}}$” where $X=V_\eta$. Suppose that $j{\!\upharpoonright\!}V:V\to V$. Then $j$ is amenable to $V$.

Suppose not and let $\alpha\in{\mathrm{OR}}$ be such that $j{\!\upharpoonright\!}V_\alpha\notin V$. Let $\beta=j(\alpha)$. Let $\widetilde{k}$ be such that $\widetilde{k}_G=j{\!\upharpoonright\!}V_{\alpha+5}$. We may assume that $${\mathbb P}{\dststile{}{}}\widetilde{k}:\check{V_{\alpha+5}}\to\check{V_{\beta+5}}\text{ is 
elementary}$$ and $${\mathbb P}{\dststile{}{}}\widetilde{k}{\!\upharpoonright\!}\check{V_\alpha}\notin\check{V}.$$ In $V[G]$, let $f:{\mathbb P}\to V_\alpha^V$ be such that $f(p)$ is some $x$ such that $$V{\models}\neg\exists y\ [ p{\dststile{}{}}_{\mathbb P}\widetilde{k}(\check{x})=\check{y}];$$ this exists by $X$-${\mathsf{AC}}$ in $V[G]$. Let $\widetilde{f}\in V$ be such that $\widetilde{f}_G=f$ and $$V{\models}{\mathbb P}{\dststile{}{}}\widetilde{f}:\check{{\mathbb P}}\to\check{V_\alpha}.$$

In $V$, define the partial function $f_*:_{\mathrm{p}}{\mathbb P}{\times}{\mathbb P}\to 
V_\alpha$ by $$f_*(p,q)=\text{ unique }x\text{ such that 
}q{\dststile{}{}}_{\mathbb P}\widetilde{f}(\check{p})=\check{x},$$ if $q\leq p$ and there is such an $x\in V_\alpha$, and $f^*(p,q)$ is undefined otherwise. Let $D={\mathrm{dom}}(f_*)$.

Let $g_*=j(f_*)\in V$ and let $p_0\in{\mathbb P}$ be such that $$V{\models}p_0{\dststile{}{}}_{\mathbb P}\widetilde{k}(\check{f_*})=\check{g}_*.$$ Since ${\mathbb P}\in V_{\kappa}$ and by elementarity, $g_*:D\to V_{j(\alpha)} \text{ and } j{\circ}f_*=g_*$. Let $p_1\leq p_0$ be such that $$V{\models}p_1{\dststile{}{}}_{\mathbb P}\widetilde{k}{\circ}\check{f_*}=\check{g_*}.$$ Since ${\mathbb P}{\dststile{}{}}\widetilde{k}\notin\check{V}$, we may fix $q\leq p_1$ with $(p_1,q)\in D$. Let $x=f_*(p_1,q)$. So by choice of $f,x$, $$V{\models}\neg\exists y\ [p_1{\dststile{}{}}_{\mathbb P}\widetilde{k}(\check{x})=\check{y}].$$ But considering the line above, and letting $y=g_*(p_1,q)$ we have $$V{\models}p_1{\dststile{}{}}_{\mathbb P}\widetilde{k}(\check{x})=\check{y},$$ a contradiction.

The assumption of $X$-${\mathsf{AC}}$ above is not fully necessary. Note that a similar proof works if $V[G]{\models}$“$(X,V_\beta)$-${\mathsf{AC}}$” for some $\beta<\kappa$, where this asserts that for each function $f$ with domain $X$, there is a $V_\beta$-choice function for $f$; that is, there is a partial function $g$ with domain ${\subseteq}X{\times}V_\beta$, such that for each $x\in X$, there is $y\in V_\beta$ with $(x,y)\in{\mathrm{dom}}(g)$, and for each $(x,y)\in{\mathrm{dom}}(g)$, we have $g(x,y)\in f(x)$.

\[tm:Reinhardt\_with\_enough\_AC\_restricts\_to\_ground\] Let $V{\subseteq}V[G]$ be a set-forcing extension via ${\mathbb P}{\subseteq}V_\eta^V$. Suppose that:

1.  $(V[G],j){\models}{\mathrm{ZFR}}$,

2.  $V[G]{\models}$“$V_\eta$ is wellordered and $\gamma^+$-${\mathsf{DC}}$ holds where $\operatorname{card}(V_\eta)=\gamma$”.

Then $j$ is amenable to $V$ and $(V,j{\!\upharpoonright\!}V){\models}{\mathrm{ZFR}}$.

By \[tm:enough\_AC\_ground\_def\], $V$ is definable over $V[G]$ from the parameter $V_{\eta+1}^V$. So by the elementarity of $j$, $j:V\to V$. So by \[tm:Reinhardt\_with\_enough\_AC\_restricts\], $j$ is amenable to $V$. And $(V,j{\!\upharpoonright\!}V)$ satisfies the ${\mathsf{ZF}}$ axioms because $(V[G],j)$ does and $(V,j{\!\upharpoonright\!}V)$ is definable from parameters over $V[G]$.

As a corollary, one cannot force a “new” Reinhardt-with-$\lambda$-${\mathsf{DC}}$ cardinal $\kappa$ with forcing of size ${<\kappa}$.

\[tm:no\_new\_sR\_or\_tR\] Assume $(V,P){\models}{\mathsf{ZF}}_2$. Let $G$ be $(V,{\mathbb P})$-generic for some ${\mathbb P}\in V$. Then:

1.  \[item:sR\] If $\kappa$ is super-Reinhardt in $(V[G],P[G])$ then there is $\kappa'\in{\mathrm{OR}}^V$ which is super-Reinhardt for $(V,P)$.

2.  \[item:tR\] If ${\mathbb P}\in V_\delta$ and $(V_\delta^{V[G]},V_{\delta+1}^{V[G]})$ is total Reinhardt, then $(V_\delta,V_{\delta+1})$ is total Reinhardt.

Part \[item:tR\]: Let $\delta\in{\mathrm{OR}}$ with ${\mathbb P}\in V_\delta$ and $G$ be $(V,{\mathbb P})$-generic and suppose that $(V_\delta^{V[G]},V_{\delta+1}^{V[G]})$ is total Reinhardt. In $V$, fix $A{\subseteq}V_\delta$. In $V[G]$, let $\kappa<\delta$ be $({<\delta},V,A)$-reflecting. Let $j:V_\delta\to V_\delta$ be $(V,A)$-cohering with ${\mathrm{cr}}(j)=\kappa$. We claim that $j\in V$, and hence, $j$ witnesses what we need in $V$.

It suffices to see that for each $\alpha<\delta$, we have $j{\!\upharpoonright\!}V_\alpha\in 
V$, because we have a name $\dot{j}\in V$ for $j$, and $V_\delta$ is inaccessible, which gives $j\in V$ as usual.

So fix $\alpha<\delta$ and $\beta=j(\alpha)$ and let $\dot{k}\in V_\delta$ be such that $\dot{k}_G=j{\!\upharpoonright\!}V_\alpha$. We may assume $\alpha>\kappa$. We may assume that ${\mathbb P}$ forces “$\dot{k}:V_\alpha\to V_\beta$ is elementary”. Let $\xi\in(\alpha+{\omega},\delta)$ with $\dot{k}\in V_\xi$. Note that the forcing facts in the following claim are defined from the relevant parameters over $V_\xi$:

There are $Y,\sigma,\eta\in V$, with $Y{\subseteq}V_\alpha$ and such that (i) for each $p\in{\mathbb P}$, if there is $x\in V_\alpha$ such that $p$ does not decide (with respect to ${\mathbb P}$) the value of $\dot{k}(\check{x})$ then there is $x\in Y$ with this property, (ii) $\eta<\kappa$, and (iii) $\sigma:V_\eta\to Y$ is a surjection.

(The first part here is motivated by Usuba’s discussion of Löwenheim-Skolem cardinals in [@usuba_ls].) In $V[G]$, there is some $X{\preccurlyeq}V_\xi$[^8] with $R,{\mathbb P},\dot{k}\in X$ and such that the transitive collapse $\bar{X}$ of $X$ is in $V_\kappa^{V[G]}$. For in $V[G]$, let $$\ell:V_\delta^{V[G]}\to V_\delta^{V[G]}$$ be elementary with ${\mathrm{cr}}(\ell)=\kappa$ and $\ell(\kappa)>\xi$ and $\ell(V)=V$. Then $$\ell``V_\xi{\preccurlyeq}\ell(V_\xi)=V_{\ell(\xi)},$$ and $\ell``V_\xi\in V_\delta[G]$, and $\ell``V_\xi$ has transitive collapse $V_\xi\in V_{j(\kappa)}^{V[G]}$. So $V_\delta[G]{\models}$“There is $X{\preccurlyeq}V_{\ell(\xi)}$ with $\ell(R,{\mathbb P},\dot{k})\in X$ and such that the transitive collapse $\bar{X}$ of $X$ is in $V_{\ell(\kappa)}^{V[G]}$”. So pulling back, in $V[G]$ we get an $X{\preccurlyeq}V_{\xi}$ with $R,{\mathbb P},\dot{k}\in X$ and such that the transitive collapse $\bar{X}$ of $X$ is in $V_\kappa^{V[G]}$. Let $\dot{\bar{X}}\in V$ be a name for $\bar{X}$; we may take $\dot{\bar{X}}\in V_\kappa$. Let $\pi:\bar{X}\to X$ be the uncollapse map. Let $\dot{\pi}\in V$ be a name for $\pi$.

Now we may assume that ${\mathbb P}$ forces “$\dot{\pi}:\dot{\bar{X}}\to\check{V_\xi}$”. Using the forcing relation in $V$, looking at all outputs forceable for $\dot{\pi}$, we can find some $Y{\subseteq}V_\alpha$ such that $X{\cap}V_\alpha{\subseteq}Y$ and $\eta<\kappa$ and a surjection $\sigma:V_\eta\to Y$ which works.

Let $\tau=j(\sigma)\in V$, so $\tau=j{\circ}\sigma$. Let $p\in G$ force “$\dot{j}:\check{V_\delta}\to\check{V_\delta}$ is elementary with ${\mathrm{cr}}(\dot{j})=\check{\kappa}$ and $\dot{j}{\!\upharpoonright\!}\check{V_\alpha}=\dot{k}$ and $\check{\tau}=\dot{j}(\check{\sigma})=\dot{j}{\circ}\check{\sigma}$”. Note that $p$ forces the value of $\dot{k}(\check{x})=\dot{j}(\check{x})$ for every $x\in{\mathrm{rg}}(\sigma)$. But then by the claim, $p$ forces the value of $\dot{k}(\check{x})$ for every $x\in V_\alpha$. Therefore $j{\!\upharpoonright\!}V_\alpha\in V$, as desired.

Part \[item:sR\]: Let $V[G]$ be a set-generic extension such that $\kappa$ is super-Reinhardt in $V[G]$. Note then that if $j:V[G]\to V[G]$ is elementary with ${\mathrm{cr}}(j)=\kappa$, then $j(\kappa)$ is also super-Reinhardt in $V[G]$ (consider $j(k)$ for various $k$). And Reinhardt easily implies supercompact. So (see [@usuba_ls]) $V[G]{\models}$“There is a proper class of supercompact cardinals”, so $V[G]{\models}$“There is a proper class of Löwenheim-Skolem cardinals”, so all set-grounds of $V[G]$ are definable from parameters over $V[G]$. Taking $\kappa'$ super-Reinhardt then above the defining parameter, and $j':V[G]\to V[G]$ with ${\mathrm{cr}}(j')=\kappa'$, it follows that $j'{\!\upharpoonright\!}V:V\to V$. But then essentially the same argument as above shows that $j'$ is amenable to $V$, as required.

Generic $j:V[G]\to M{\subseteq}V$ {#sec:generic_j:V[G]_to_M}
=================================

Woodin proved the following theorem:

Assume ${\mathsf{ZFC}}$. Let $G$ be set-generic over $V$. Then there is no elementary $j:V[G]\to V$.

Of course in the case that $G\in V$, this reduces to Kunen’s inconsistency. We generalize this theorem below by dropping the assumption of ${\mathsf{AC}}$ in $V$, but adding the requirement that $G$ adds a set of ordinals to $V$ (which of course is true in the ${\mathsf{ZFC}}$ context, if $G\notin V$). The proof also gives an alternate proof of Woodin’s result.

\[lem:no\_gen\_emb\_into\_V\_with\_new\_O\_set\] Assume ${\mathsf{ZF}}$. Let $G$ be $(V,{\mathbb P})$-generic for some forcing ${\mathbb P}\in V$ and let $\alpha\in{\mathrm{OR}}$ and suppose $X\in{\mathcal{P}}(\alpha){\cap}V[G]{\backslash}V$. Let $\theta\in{\mathrm{OR}}$ be such that $V$ has no surjection ${\mathbb P}^{<{\omega}}{\times}\alpha\to\theta$. Then there is no $(M,E,N,j)$ such that $M,E\in V$ and $N,j\in V[G]$, $N$ is transitive and rudimentarily closed, ${\mathcal{H}}_\theta^V\cup\{X\}{\subseteq}N$, and $$j:(N,\in)\to(M,E)$$ is elementary. (Note we allow $(M,E)$ to be illfounded.)

We will prove the following claim, without the contradictory assumption that $X\in N$.

\[clm:j\_rest\_alpha\_in\_V\]$j{\!\upharpoonright\!}\alpha\in V$.

Assuming this claim, and adding the assumption that $X\in N$, we can easily deduce that $X\in V$, a contradiction, completing the proof. For each $\xi<\alpha$, we have $$\xi\in X\iff N{\models}\text{``}\xi\in X\text{''}\iff
(M,E){\models}\text{``}j(\xi)\in j(X)\text{''},$$ but since $(M,E)\in V$ and $j{\!\upharpoonright\!}\alpha\in V$, therefore $X\in V$.

We prove some sub-claims:

\[clm:no\_surjection\_to\_theta\] $V[G]{\models}$“there is no surjection $f:{\mathbb P}^{<{\omega}}{\times}\alpha\to\theta$”.

Suppose $f\in V[G]$ is such a surjection, and let $\widetilde{f}\in V$ be such that $\widetilde{f}_G=f$. Let $p_0\in G$ be such that $$V{\models}p_0{\dststile{}{}}_{\mathbb P}\text{``}\widetilde{f}:\check{{\mathbb P}}^{<{\omega}}{\times}\check{\alpha} 
\to\check { \theta} \text{ is a surjection''}.$$ Work in $V$. We define a function $g$. For $(p,\vec{q},\xi,\gamma)\in{\mathbb P}{\times}{\mathbb P}^{<{\omega}}{\times}\alpha{\times}\theta$, set $$g(p,\vec{q},\xi)=\gamma\iff 
p\leq 
p_0\text{ 
and }p{\dststile{}{}}_{\mathbb P}\widetilde{f}(\check{\vec{q}},\check{\xi})=\check{\gamma}.$$ Set $A={\mathrm{dom}}(g){\subseteq}{\mathbb P}{\times}{\mathbb P}^{<{\omega}}{\times}\alpha$. Clearly $g:A\to\theta$ is surjective, which contradicts our choice of $\theta$.

Now work in $V[G]$. We have $j{\!\upharpoonright\!}\theta:\theta\to{\mathrm{OR}}^M$. Let $\widetilde{k}\in V$ be such that $\widetilde{k}_G=j{\!\upharpoonright\!}\theta$. Let $p_0\in G$ be such that $$V{\models}p_0{\dststile{}{}}_{\mathbb P}\text{``}\widetilde{k}:\check{\theta}\to{\mathrm{OR}}^{\check{M}}\text{''}.$$ (Here ${\mathrm{OR}}^M=\{x\in M\bigm|(M,E){\models}\text{``}x\text{ is an ordinal''}\}$; we do not assume that $M$ is wellfounded.)

Working in $V$, for $p\in{\mathbb P}$ with $p\leq p_0$, let $$X_p=\{\xi<\theta\bigm|\exists x\in {\mathrm{OR}}^M\ 
[p{\dststile{}{}}\widetilde{k}(\check{\xi})=\check{x}]\}.$$ Note that for each such $p$, we have $X_p\in V$ and $j_p=j{\!\upharpoonright\!}X_p\in V$.

\[clm:X\_p\_long\] There is $p\in G$ such that $p\leq p_0$ and $X_p$ has ordertype $\beta>\alpha$.[^9]

We have $\theta=\bigcup_{p\in G, p\leq p_0}X_p$. Now suppose the claim fails. Then working in $V[G]$, we may define a surjection $$f:{\mathbb P}{\times}\alpha\to\theta$$ by setting $f(p,\xi)=$ the $\xi$th element of $X_p$, in the case that $X_p$ is defined and has ordertype $>\xi$, and $f(p,\xi)=0$ otherwise. This contradicts Claim \[clm:no\_surjection\_to\_theta\].

So fix $p$ as in Claim \[clm:X\_p\_long\]. Let $\alpha'$ be the $\alpha$th element of $X_p$. So $Y=X_p{\cap}\alpha'$ has ordertype $\alpha$. Let $\pi:\alpha\to Y$ be the increasing enumeration of $Y$. Then $\alpha,Y,\pi\in{\mathcal{H}}_\theta^V{\subseteq}N$. Since $N$ is transitive, $$N{\models}\text{``}\pi:\alpha\to Y\text{ is the increasing enumeration of  }
Y\text{''},$$ so applying $j$, $$(M,E){\models}\text{``}j(\pi):j(\alpha)\to j(Y)\text{ is
 the increasing enumeration
of }j(Y)\text{''}.$$

Let $j_Y=j{\!\upharpoonright\!}Y\in V$. Note that $j_Y(\pi(\xi))=j(\pi(\xi))=j(\pi)(j(\xi))$ for each $\xi<\alpha$,[^10] and so[^11] $$j{\!\upharpoonright\!}\alpha=j(\pi)^{-1}{\circ}j_Y{\circ}\pi,$$ and since $j_Y,\pi,j(\pi)\in V$ and $(M,E)\in V$, we get $j{\!\upharpoonright\!}\alpha\in 
V$, as desired.

With Claim \[clm:j\_rest\_alpha\_in\_V\] established, the theorem follows as explained above.

From the lemma, we can deduce:

\[tm:no\_gen\_emb\_into\_V\_with\_new\_O\_set\] Assume $(V,P){\models}{\mathsf{ZF}}_2$. Let $G$ be $(V,{\mathbb P})$-generic for some ${\mathbb P}\in V$. Suppose there are $\alpha\in{\mathrm{OR}}$ and $X{\subseteq}\alpha$ with $X\in V[G]{\backslash}V$.

Then there is no $(j,M)\in P[G]$ with $M{\subseteq}V$ and $j:V[G]\to M$ elementary. Hence, there is no $j\in P[G]$ such that $j:V[G]\to V$ is elementary.

Question: Is it consistent that there is $j:V[G]\to M{\subseteq}V$ as in the statement of Theorem \[tm:no\_gen\_emb\_into\_V\_with\_new\_O\_set\], if we instead have ${\mathcal{P}}(\alpha){\cap}V[G]{\subseteq}V$ for each $\alpha\in{\mathrm{OR}}$? Does it follow that $V^{<{\mathrm{OR}}}{\cap}V[G]{\subseteq}V$? What about in the case that $M=V$?

[^1]: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure.

[^2]: The author thanks Andreas Lietz for pointing this fact out to him.

[^3]: See \[dfn:codcof\] and Lemma \[lem:reg\_non\_inacc\].

[^4]: \*\*\*This theorem has now been superseded by the next one, which reduces the inaccessibility assumption to $\operatorname{cof}(\delta)>{\omega}$. But it’s easier to start with the proof of this one, and I will probably integrate some details from its proof into the better one. Probably again some details are unnecessary since there is the general development of ultrapowers now.

[^5]: \[ftn:mof\]For the discussion see https://mathoverflow.net/questions/185253/reinhardt-cardinals-and-iterability. Note there is a slight difference in notation; we are writing $N_\alpha$ for what he refers to as $M_\alpha$ there.

[^6]: Note that the elementarity is with respect to $M_\alpha=(N_\alpha,j_\alpha)$ and $M_\beta=(N_\beta,j_\beta)$, not just $N_\alpha,N_\beta$.

[^7]: Of course, in $N_{\omega}[G]$ we can compute the relevant truth of $N_{\omega}$, either using the definability of set-grounds, or simpler, just noting that whenever $V_\xi{\preccurlyeq}_n V$, then $V^{N_{\omega}}_{i_{0{\omega}}(\xi)}{\preccurlyeq}_n N_{\omega}$, and of course $V_{i_{0{\omega}}(\xi)}^{N_{\omega}}\in N_{\omega}[G]$.

[^8]: We write $V_\xi$ for rank segments of $V$ and $V_\xi^{V[G]}$ for those of $V[G]$.

[^9]: Note that the proof does not use ${\mathsf{AC}}$.

[^10]: Here $j(\pi)(j(\xi))$ is a slight abuse of notation; it should really be $(j(\pi)(j(\xi)))^M$, i.e. the $x\in M$ such that $M{\models}$“$x=j(\pi)(j(\xi))$”.

[^11]: The notation “$j(\pi)^{-1}$” involves a similar abuse.
