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Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ is true (and some in which $CH$ is not a relevant hypothesis?).

In formulating logical languages, there has been an outstanding problem: that of deciding the Law of the Excluded Middle $p\vee\neg p$. In classical logic $p\vee\neg p$ is true, but in intuitionistic logic this is not the case. With $CH$, the pragmatic mathematician tries to avoid invoking $CH$; if he assumes $CH$ or $\neg CH$, he will state it clearly. In everyday mathematics practice, the mathematician does use $p\vee\neg p$, but we do see an effort to give effective constructions, and hard estimates in analysis.

It seems that people are using classical logic basically because it's a core logic, in analogy to the constructible universe $L$ in which $CH$ is true. It is the logic first discovered (and often wrongly attributed as the Platonic choice) and the consistency and strength of other languages proven in terms of this core logic. Afterwards people come up with other logics, like Brouwer coming up with intuitionistic logic, in which a fundamental principle, the Law of the Excluded Middle, does not hold. It seems to me that this debate regarding the Law of Excluded Middle can be formalized using the multiverse approach.

So has anyone tried to use the multiverse approach towards considering these plurality of languages? Perhaps by considering a multiverse of topoi?

PS. The difficulty for arithemetizable syntax to describe continuous properties of geometric/measure-theoretic space is central (may I say?) to the difficulty of deciding $CH$. In a like fashion, the difference between classical and intuitionistic logic plays up in comparing the Dedekind-reals and the Cauchy-reals. In other words, continuous properties have different descriptions in these two logics. I'm hoping that work on the Law of the Excluded Middle will shed some light on trying to use discrete languages to model continuous properties. There's more where this comes from, but it is enough for now...

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I don't understand your remark about the reals. Whether Dedekind reals and Cauchy reals are isomorphic depends rather on what kind of choice is available, not whether the ambient logic is classical. More precisely, if countable choice is valid, then the Dedekind and the Cauchy reals coincide, irrespective of the Law of Excluded Middle. Can you please clarify what you meant? –  Andrej Bauer May 19 '10 at 13:39
And another point: the 2-category of toposes and geometric morphisms (what I think you call "multiverse") has been studied extensively and in some depth. –  Andrej Bauer May 19 '10 at 13:41
I am not sure that I agree with your remark that "the pragmatic mathematician tries to avoid invoking $CH$." It seems to me that even pragmatic mathematicians are aware of the fact that if you invoke $CH$, you have only proved a consistency statement. On the other hand, I once had a referee's report which claimed that any non-logician who accepts $AC$ also accepts $CH$ ... –  Simon Thomas May 19 '10 at 13:51
@Bauer It is my ignorance about Dedekind and the Cauchy reals. I'll have a look at the 2-category of toposes. Thanks. –  Colin Tan May 19 '10 at 13:58
Simon, the converse of your referee's view is probably true: mathematicians who accept CH almost universally also accept AC. (In the extreme form, anyway, we know GCH implies AC.) Colin, I like your question, but my own multiverse view is unabashedly classical, so I have little to say. But I'd like to hear from the intuistionists. As Andrej says, isn't topoi theory already a multiverse theory of the type you seek? –  Joel David Hamkins May 19 '10 at 16:26

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I am not sure I understand all remarks that Colin made, and I disagree with some of them, but I can comment on the idea of "multiverse". Let us consider the following positions:

  1. There is a standard mathematical universe.
  2. There are many mathematical universes.
  3. There is a multiverse of mathematics.

These are supposed to be informal statements. You may interpret "universe" as "model of set theory ZF(C)" or "topos" if you like, but I don't want to fix a particular interpretation.

I would call the first position "naive absolutism". While many mathematicians subscribe to it on some level, logicians have known for a long time that this is not a very useful thing to do. For example, someone who seriously believes that there is just one mathematical universe would reject model theory as fiction, or at least brand it as a technical device without real mathematical content. Such opinions can actually slow down mathematical progress, as is historically witnessed by the obstruction of "Euclidean absolutism" in the development of geometry.

The second position is what you get if you take model theory seriously. Set theorists have produced many different kinds of models of set theory. Why should we pretend that one of them is the best one? You might be tempted to say that "the best mathematical universe is the one I am in" but this leads to an unbearably subjective position and strange questions such as "how do you know which one you are in?" At any rate, it is boring to stay in one universe all the time, so I don't understand why some people want to stick to having just one cozy universe.

I need to explain the difference between the second and the third position. By "multiverse" I mean an ambient of universes which form a structure, rather than just a bare collection of separate universes. The difference between studying a "collection of universes" and studying a "multiverse" is roughly the same as the difference between Euclid's geometry and Erlangen program -- both study points and lines but conceptual understanding is at different levels. Likewise, a meta-mathematician might prove interesting theorems about models of set theory, or he might consider the overall structure of set-theoretic models.

It should be obvious at this point that there cannot be just one notion of "multiverse". I can think of at least two:

  • Set theory studies the multiverse of models of ZF. The structure is studied via notions of forcing, and probably other things I am not aware of.
  • Topos theory studies the multiverse of toposes. The structure of the multiverse is expressed as a 2-category of toposes (and geometric morphisms).

I do not mean to belittle set theory, but in a certain sense topos theory is more advanced than set theory because it uses algebraic methods to study the multiverse (I consider category theory to be an extension of algebra). In this sense the formulation of forcing in terms of complete Boolean algebras by Scott and Solovay was a step in the right direction because it brought set theory closer to algebra. Set theorists should learn from topos theorists that transformations between set-theoretic models are far more interesting than the models themselves.

In the present context the question "classical or intuitionistic logic" becomes "what kind of multiverse". If multiverse is "an ambient of universes, each of which supports the development of mathematics" then taking our multiverse to be either too small or to big will cause trouble:

  • if the multiverse is too small, we will be puzzled by its ad hoc properties and we will look in vain for overall structure (imagine doing analysis with only rational numbers),
  • if the multiverse is too big, its overall structure will be poor and it will include universes whose internal mathematics is too far removed from our own mathematical experience (imagine doing analysis on arbitrary rings--I am sure it's possible but it's unlike classical analysis).

Topos theory gains little by restricting to Boolean toposes. I have never heard a topos-theorist say "I wish all toposes were Boolean". Also, toposes occurring "in nature" (sheaves on a site) typically are not Boolean, which speaks in favor of intuitionistic mathematics.

An example of an ad hoc property in too small a multiverse occurs in set theory. We construct models of set theory by forming Boolean-valued sets which are then quotiented by an ultrafilter. What is the ultrafilter quotient for? The algebraic properties of Boolean-valued sets are hardly improved when we pass to the quotient, not to mention that it stands no chance of having an explicit description. A possible explanation is this: we are looking only at one part of the set-theoretic multiverse, namely the part encompassed by Tarskian model theory. Our limited view makes us think that the ultraquotient is a necessity, but the construction of Boolean-valued models exposes the ultraquotient as a combination of two standard operations (product followed by a quotient). We draw the natural conclusion: a model of classical first-order theory should be a structure that measures validity of sentences in a general complete Boolean algebra. The Boolean algebra $\lbrace 0,1 \rbrace$ must give up its primacy. What shall we gain? Presumably a more reasonable overall structure. At first sight I can tell that it will be easy to form products of models, and that these products will have the standard universal property (contrast this with ultraproducts which lack a reasonable universal property because they are a combination of a categorical limit and colimit). Of course, there must be much more.

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Thank you Andrej for your response. It clarified much. –  Colin Tan May 20 '10 at 15:33
+1 Another very nice post Andrej! Being a person who walks both sides of the divide, I understand your criticism of set theory but I disagree with some. Forcing is easily understandable via sheaves on a locale. Since set theory is a classical theory, set theorists always use the double-negation topology, which immediately leads to a complete Boolean algebra instead of a complete Heyting algebra. This is restrictive but it is well justified by context. It is however shameful that set theorists are generally unaware of this connection. –  François G. Dorais May 20 '10 at 16:07
The quotient by the generic filter is only a matter of convenience. Locales with the double-negation topology generally have no points. The generic filter G can be understood as "fictitious point" of the forcing locale P. This can be understood as a "geometric morphism" Set[G] -> Sh(P). In theory all forcing could be done in Sh(P), but in practice set theorists generally alternate doing computations in Set[G] and computations in Sh(P). The same occurs in other branches of mathematics, compare my answer with Emerton's here: mathoverflow.net/questions/23264/are-c-and-barq-p-isomorphic –  François G. Dorais May 20 '10 at 16:08
Why should the existence of "just one mathematical universe" entail the rejection of model theory as a fiction? –  Marc Alcobé García Jun 23 '10 at 5:57
@Andrej Bauer "For example, someone who seriously believes that there is just one mathematical universe would reject model theory as fiction, or at least brand it as a technical device without real mathematical content." This is an absurd statement. Clearly model theory has a lot of interesting content, which can be appreciated by someone who works within ZFC and believes in a single background universe. This is like saying you can't appreciate general field theory if you believe the real numbers are a single well-defined structure. Any student of field theory will see this. –  Monroe Eskew Sep 1 '11 at 19:25

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