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I read with interest both Hamkins Multiverse Axioms and Joyal and Moerdijk's algebraic set theory. Both of these perspectives takes a set-theory universe as an object, and consider collections of set-theory universes. The former lives in the setting of first-order logic, while the later lives in the setting of category theory (specifically monads and algebras).

Has there been an attempt to compare these theories? In particular (I hope the following is a well-founded question), does Hamkins' Multiverse axiomatize (the set-theory part, i.e. ignoring the intuitionistic and topos theoretic part) the class category of ZF-algebras?

I'm not too familiar with both these theories, but I have a great interest in Foundations. I would be thankful for more detailed comparisons beyond the particular question that I ask.

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I'm curious: what's a class category? – Tom Leinster Jan 22 '12 at 16:54
@Tom: Probably, in this instance, the (super)large category of categories of classes as described by algebraic set theory. @Colin, can't one write down first order axioms for a category of classes (though you need to replace talk of 'set of small arrows' by a predicate 'is a small arrow')? – David Roberts Jan 23 '12 at 0:37
@Tom: From Awodey's introduction to algebraic set theory, a class category is a category whose objects are classes. The morphisms are functions between classes, but there are special "small" functions which are called maps. – user2529 Jan 23 '12 at 2:50
Ah, a category of classes then. This was my first guess, but it seems at odds with the phrase 'category of ZF algebras'. I would have thought that a ZF-algebra was a category of sets. – David Roberts Jan 23 '12 at 3:05
@David: It seems that a ZF-algebra is not the usual algebra over a monad. A ZF-algebra is a class. The initial ZF-algebra is a model of ZF. The category of ZF algebras is thus a "category of classes", as you call it. – user2529 Jan 24 '12 at 12:53
up vote 2 down vote accepted

I think the two theories should be regarderd as existing at different levels. A category of classes, in the sense of algebraic set theory, is not a collection of models of set theory, but (an abstraction of) the collection of all classes relative to one model of set theory. In particular, if $V$ is a model of set theory, then the collection of all classes in V is a category of classes.

One can then, if one wants, define a notion of "internal model of set theory" inside a category of classes, and in the "canonical" example this would just be the notion of a class-model of set theory relative to $V$. The model $V$ itself is recoverable as the "initial" such internal model. Joel's excellent answer explains why such a collection of "internal models" will not generally satisfy the multiverse axioms.

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Thank you Mike. I think it's a really pity that there is still not a (suitable) category or moduli of set theoretic universes. Or more generaly, fix a first-order theory $T$. It would be great to have a category of moduli of models of $T$. – user2529 Jan 23 '12 at 14:35
I think the 2-category of toposes is a better candidate for the category of set theoretic universes. It probably doesn't satisfy all the multiverse axioms either, but at least it approximates some of them, e.g. closure under forcing. – Mike Shulman Jan 23 '12 at 17:00

My understanding of algebraic set theory is imperfect, but I am given to understand that what is involved is some kind of category consisting of class models of set theory. To formalize it, I would want to ask the question in the context of a background set theory, such as GBC or some other second-order set theory, where one has a universe $V$ of ZFC, and the corresponding family of classes of $V$, which satisfy the GBC axioms of set theory. Let me assume that the intended category is included within the family of these classes that are models of some fragment of ZFC, that is, classes of the form $(W,E)$, where $W\subset V$ is a class of objects and $E\subset W\times W$ is a binary relation on $W$ satisfying the ZFC axioms. And furthermore, allow me to assume that $(V,\in)$ itself is supposed to be one of the models in the intended category.

If this understanding of the situation is correct (and please correct me if it is not), then there will be problems with several of the multiverse axioms, and they will not all hold in this setting. Let me mention a few of the problems.

First, the forcing extension axiom will fail (see my paper The set-theoretic multiverse for the axioms). This is the multiverse axiom asserting that if $W$ is any universe and $\mathbb{P}\in W$ is a forcing notion in $W$, then there will be a corresponding $W$-generic forcing extension $W[G]$ with $G\subset\mathbb{P}$. If $V$ itself is one of the universes, however, then there will have to be classes in $V$ that represent forcing extensions $V[G]$ by nontrivial forcing notions $G\subset\mathbb{P}\in V$. But this is impossible, since no such generic filters $G$ can exist in $V$, and therefore no such class can exist in $V$. (There is a more subtle sense in which one can have such classes, if one uses the Boolean ultrapower understanding of forcing, by which one has an elementary embedding $j:V\to\overline{V}$ such that in $V$ there is a $\overline{V}$-generic filter $G\subset j(\mathbb{P})$, but this is not quite the same thing.)

The previous argument assumed that $V$ was one of the universes in the intended category. One could weaken this just to the assertion that there was some universe $W$ in the category that had the true $\omega$ and also some uncountable sets in it. In this case, the forcing to collapse those sets to $\omega$ would be a forcing notion in $W$, but could have no $W$-generic filter in $V$, since any such filter would really collapse those sets to become countable, which they are not in $V$, and so there would be none of the required forcing extensions of $W$ in the category. Basically, the forcing extension axiom asserts that one must really reach outside the current set-theoretic background to achieve it, and so if one builds the category of universes inside the set-theoretic background, one will not have the generic objects available.

Similar reasoning shows that many of the other multiverse axioms will fail for this category. For example, some of the multiverse axioms imply that every universe is a countable transitive model inside another universe of the multiverse. But again, if the category of universes includes any actually uncountable transitive universes, then this is impossible inside any class model of $V$.

Further problems arise for such a category of universes with the well-founded mirage axiom, which asserts that every universe is seen as having a non-standard $\omega$ by another universe in the multiverse (one of the more controversial axioms). This axiom will fail if the category of universes is to include any model with what is to the $V$ perspective the standard $\omega$, since $V$ can have no class model that looks upon the true $\omega$ of $V$ as ill-founded, since then $V$ itself would look upon its own $\omega$ as ill-founded.

Meanwhile, in my paper A natural model of the multiverse axioms with Victoria Gitman, we prove that if ZFC is consistent, then the collection of computably-saturated countable models of ZFC is a model of all the multiverse axioms (and furthermore any collection of models of set theory satisfying the multiverse axioms must consist entirely of computably saturated models). Many of our arguments in that paper involve the kind of algebraic treatment of models of set theory that I naively expect to be a part of the analysis in algebraic set theory, and so my expectation is that if there are bridges to be built or discovered between the two perspectives that you mention, I would expect to find them there, in the realm of collections of highly saturated models of set theory.

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I think your understanding is close, but not quite right: the objects of a category of classes are just classes, without the extra structure of a membership relation satisfying any axioms. But one can of course consider objects of a category of classes equipped with such structure, and then I think your answer applies to such things. – Mike Shulman Jan 23 '12 at 6:59
Ah, it seems from your answer that the idea is closer to a categorification of second order set theory. So the presumption of my answer that we had a category of models of set theory is off base. – Joel David Hamkins Jan 23 '12 at 11:49
Vote this comment up if I should delete my answer. – Joel David Hamkins Jan 23 '12 at 12:15
Keep the answer around! There is still a lot of valuable information here. As a general rule, I think that keeping all (nonspam) answers is a good idea. It gives a record of thoughts in process, which is a VERY rare thing to find in mathematics. Mathoverflow is one of the few places in the world where you can read the real time thought process of a mathematician, not a polished distillation of that process. – Steven Gubkin Jan 23 '12 at 16:03
mbsq, to continue: in particular, my view is that technical developments analogous to forcing will eventually arise, by which we shall be able to construct new set-theoretic universes allowing us to modify arithmetic truth with the same ease and flexibility with which we may currently modify higher order truth via forcing; this situation will undermine the current widespread confidence that there is an absolute notion of the finite, just as our experience with forcing has undermined confidence in absolute set-theoretic truth. – Joel David Hamkins Jan 23 '12 at 19:19

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