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Mathias forcing consists of conditions of the form $(s,A)$ where $s$ is a finite subset of $\omega$, $A$ is an infinite subset of $\omega$ such that $\max(s) < \min(A)$ and $(s,A)\leq (t,B)$ iff $s\supseteq t$, $A\subseteq B$ and $s\setminus t\subseteq B$.

Laver's forcing, on the other hand, consists of conditions $p$ that are subtrees of $\omega^{< \omega}$ (i.e. they are subsets closed under initial segments) such that there is a node $s\in p$, called the stem of $p$, that is comparable with every element of $p$ and moreover, for every $t\in p$, if $s\subseteq t$ then {$n < \omega | t\frown n\in p$} is infinite.

In both forcings, the union of the "working" parts (i.e. the first coordinate in the case of Mathias, the stem in the case of Laver) of the elements of the generic filter give us a generic real. My question is whether there is an easy-to-grasp difference between the generic models that one can obtain by means of these two forcings. This is motivated by the (empirical) "fact" that, based on my experience, everything that can be done with Mathias forcing can usually also be done with Laver's (which is often much more intuitive). For example, both forcings "kill" uncountable strong measure zero sets from the ground model. Also, both forcings are proper, and the proofs for each of the forcings are fairly similar. I sort of understand why they are not forcing equivalent (well, at least I understand why my attempts to define a dense embedding of one into the other fail), and I know that if you iterate these forcings you can actually see some differences. For example, if you look at the table of cardinal characteristics of the continuum at the end of Blass's article in the Handbook of Set Theory, you can see that e.g. the splitting number $\mathfrak s$ attains a value of $\omega_1$ in Laver's model, as opposed to a value of $\mathfrak c$ (which in this case is $\omega_2$) in Mathias model. However, these models are the result of iterating these forcings $\omega_2$ many times with countable support, and I'm looking for something more immediate. So, my specific question is:

Suppose $V$ is a fixed model of ZFC (plus CH, or any other hypothesis you might need), and that $G$ is a Laver-generic filter over $V$, and that $H$ is a Mathias-generic filter over $V$. What would be a (or some) easy to grasp (as much as possible, but of course if it's too technical I don't mind going through it) difference between $V[G]$ and $V[H]$? (Thus, I'm not asking for a difference between Laver's and Mathias's model (that are the result of adjoining $\omega_2$ many Laver or Mathias reals, respectively), but for a difference between the models obtained by adding a single Laver or Mathias real).

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2 Answers 2

up vote 10 down vote accepted

In the Mathias extension (extension by a single Mathias real) the collection $V\cap 2^\omega$ (the reals from the ground model) will have measure zero. This is not the case for the Laver extension.

Both facts can be found in the Bartoszynski-Judah book Set Theory: On the Structure of the Real Line. The former fact is Lemma 7.4.2 while the latter is a consequence of Theorem 7.3.39. That Mathias forcing makes $V\cap 2^\omega$ have measure zero is not unrelated to (as you pointed out) the fact that iterating Mathias forcing increases the splitting number. The Mathias real is an unsplit real: an $x\in[\omega]^\omega$ that for every $y\in V\cap[\omega]^\omega$ either $x\subseteq^*y$ or $x\cap y$ is finite. For a fixed such $x$ the collection of such $y$ has measure zero.

Another interesting difference is in the available subforcings. Like a Sacks real, a Laver real $r$ is an object of minimal degree; whenever $x\in V[r]$ if $x$ doesn't belong to the ground model then we can recover $r$ from $x$ (ie $V[r]=V[x]$.) This is a special case of Theorem 7 in Groszek's 'Combinatorics on ideals and forcing with trees'. On the other hand this fails badly for Mathias forcing. If $m\in[\omega]^\omega$ is a Mathias real, and $m_0,m_1$ are subsets of $m$ with $m_0\cap m_1$ finite, then $V[m_0]$ and $V[m_1]$ are distinct extensions. (This is Corollary 8.3 in Mathias's original paper 'Happy Families').

If you're interested in more ways in which the two are very similar you can read Section 3 of Brendle's 'Combinatorial properties of classical forcing notions' which shows that the two forcings have a similar effect on many of the cardinal characteristics in the Cichon diagram.

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Thanks Justin, this is really helpful! –  David FernandezBreton May 2 '12 at 2:36

In this paper Alan Dow compares Laver and Mathias forcing regarding their effects on the algebra $\mathcal{P}(\omega)/\mathit{fin}$. He considers four iterations of length $\omega_2$: iterate $L$, iterate $M$, iterate $L*M$ or iterate $M*L$ and calculates the values of many familiar cardinal characteristics of the continuum in the resulting models.

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