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August 23, 2011

The Set-Theoretic Multiverse

Posted by David Corfield

There’s an interesting paper out today on the ArXiv – Joel Hamkins’ The set-theoretic multiverse.

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

So set theorists’ experience of dealing with various models of set theory, and the small modifications needed to generate models either satisfying or not satisfying the continuum hypothesis, tells them that settling its truth or falsity by devising an evident axiom from which either it or its negation follows is not a live option. Gödel’s platonism was wrong then, yet Hamkins retains a form of realism:

The multiverse view is one of higher-order realism – Platonism about universes – and I defend it as a realist position asserting actual existence of the alternative set theoretic universes into which our mathematical tools have allowed us to glimpse. The multiverse view, therefore, does not reduce via proof to a brand of formalism. In particular, we may prefer some of the universes in the multiverse to others, and there is no obligation to consider them all as somehow equal.

Naturally, I looked to see the part Hamkins sees category theory playing in this multiverse. While not extensively considered in the paper, he writes

Set theory appears to have discovered an entire cosmos of set-theoretic universes, revealing a category-theoretic nature for the subject, in which the universes are connected by the forcing relation or by large cardinal embeddings in complex commutative diagrams, like constellations filling a dark night sky. (p. 3)

and

In what appears to be an interesting case of convergent evolution in the foundations of mathematics, this latter universe concept coincides almost completely with the concept of Grothendieck universe, now pervasively used in category theory [Krö01]. The only difference is that the category theorists also view V ωV_{\omega} as a Grothendieck universe, which amounts to considering 0\aleph_0 as an incipient inaccessible cardinal. Surely the rise of Grothedieck universes in category theory shares strong affinities with the multiverse view in set theory, although most set theorists find Grothendieck universes clumsy in comparison with the more flexible concept of a (transitive) model of set theory; nevertheless, the category theorists will point to the multiverse concepts present in the theory of toposes as more general still (see [Bla84]). (p. 13)

Bla84 is Andreas Blass, The interaction between category theory and set theory.

We’ve had our own expression of multiverse thinking in this thread.

Posted at August 23, 2011 3:18 PM UTC

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Re: The Set-Theoretic Multiverse

Is “multiverse” really a good term to use? Surely this will unnecessarily attract more physics/quantum mechanics crackpots than you want…

Posted by: asdf on August 23, 2011 5:44 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

You’ll have to take that up with Joel Hamkins. But your asking caused me to see that William James coined the term multiverse, something I never knew.

Posted by: David Corfield on August 23, 2011 8:12 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

The absence of a reference to Alian Badiou http://en.wikipedia.org/wiki/Alain_Badiou

who works on the subject for decades is a bit strange

Posted by: N Mnev on August 23, 2011 7:42 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

The page you mention doesn’t mention the word ‘multiverse’, so I don’t see your point. Do you mean the business about

Only set theory allows one to conceive a ‘pure doctrine of the multiple’

in this section?

I can’t say the following is too promising

Badiou’s use of set theory in this manner is not just illustrative or heuristic. Badiou uses the axioms of Zermelo–Fraenkel set theory to identify the relationship of being to history, Nature, the State, and God.

Posted by: David Corfield on August 23, 2011 8:25 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

I’ve struggled with reading one of Badious texts, I forget which. But I do recall that he talks about turning philosophy away from poetry/romanticism, and embracing mathematics again. He can write well. But the main text I found difficult and turgid.

Although he does focus on the zermelo-fraenkel axioms in the texts I’ve seen, I am aware that he’s used Topos Theory in others.

The philosophers in the seminar I went to were interested in my critique, given that I have some background in mathematics, and they mainly didn’t, of the foundational status of the zermelo-fraenkel axioms.

Posted by: Mozibur Ullah on August 24, 2011 12:06 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

Badieu has no pop-word multiverse. He has an ontological concept of World. With a contunuum conjecture forcing and topos machinery.
This mainly in his books “Logics of Worlds” and “Being and Event” He is a philosopher but he learned math good enough.

Posted by: N Mnev on August 24, 2011 5:06 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

What a great paper! I’m immensely encouraged to see at least some ZF-theorists moving towards a view of this sort; arguably, it is long overdue. (My personal philosophy tends more towards formalism than Platonism, but given that one is a Platonist, the “multiverse” view seems to me much more defensible and useful than the “universe” one.) I’m particularly happy that he extends his pluralism explicitly to “the” natural numbers, and I very much appreciate his argument for the impossibility of “settling” the continuum hypothesis:

Our situation, after all, is not merely that CH is formally independent and we have no additional knowledge about whether it is true or not. Rather, we have an informed, deep understanding of the CH and ¬CH worlds and of how to build them from each other…. Consequently, if you were to present a principle Φ and prove that it implies ¬CH, say, then we can no longer see Φ as obviously true, since to do so would negate our experiences in the set-theoretic worlds having CH. … The situation would be like having a purported ‘obviously true’ principle that implied that midtown Manhattan doesn’t exist. But I know it exists; I live there. Please come visit!

Of course, this argument could just as well be extended to the axiom of choice and the law of excluded middle. ZFC-theorists may not be as familiar with the universes in which these fail, but constructive mathematicians and topos theorists are quite familiar with them.

Along those lines, around here it seems appropriate to emphasize further the point that

category theorists will point to the multiverse concepts present in the theory of toposes as more general still

Since models of ZFC are entirely first-order equivalent to models of ETCS + Replacement, the multiverse of models of ZFC might just as well be considered a multiverse of models of ETCS+R. The latter view, of course, has the advantage that we can consider functors and transformations between models of ETCS+R, making the multiverse into a 2-category.

I’m also intrigued by his brief description of the “modal logic of forcing”, which I don’t recall having encountered before. I wonder whether this modal logic is at all related to some sort of internal logic of a multiverse 2-category.

Posted by: Mike Shulman on August 24, 2011 6:50 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

[…] making the multiverse into a 2-category.

Which then also provides a good solution to the problem that asdf sees above.

Let’s not say “multiverse”. Let’s say “2-category” of set theory models.

Posted by: Urs Schreiber on August 24, 2011 8:51 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

As I understand it, forcing conditions can be understood as a kind of Kripke model. Suppose you have a ZFC model MM, to which you wish to add a new generic element GG. So, staying in MM, you construct a poset of approximations (whose elements I will write g,hg,h) of the new generic element GG. Then, propositions ϕ\phi interpreted M[G]M[G] are translated into formulas gϕg \Vdash \phi, with the usual Kripke monotonicity condition that if ghg \leq h (i.e., gg is a more precise approximation than hh)and hϕh \Vdash \phi, then gϕg \Vdash \phi. And of course, if you have a nontrivial Kripke structure you get interesting modalities.

I’ve never used proper Cohen forcing (I only really know it via realizability models of classical logic), so this explanation might be off in some details.

Posted by: Neel Krishnaswami on August 24, 2011 9:21 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

Yes, forcing with a single set of forcing conditions is very similar to Kripke models. Both are special cases of the construction of sheaves on a site: Kripke models are presheaves on a poset, while forcing is sheaves for the double-negation topology on a poset.

However, from reading the paper, it looks like what Joel means by the “modal logic of forcing” involves rather modalities that quantify over all possible forcing conditions. He writes ϕ\diamondsuit \phi to mean “ϕ\phi holds in some forcing extension”, and “ϕ\Box \phi” to mean “ϕ\phi holds in all forcing extensions”. Have a look at section 10.

Posted by: Mike Shulman on August 24, 2011 4:59 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

The modal logic of forcing explains at greater length. The logic is S4.2, see Theorem 3.

From the quotation in my post, there are also large cardinal embeddings. I wonder if there’s a modality for large cardinal embeddings. Are there any other kinds of arrow between universes?

Why do we not introduce modal concepts in other categories? E.g., in the category of groups one could say a property is necessarily held by a group if it is held at all codomains of arrows whose domain is that group.

Posted by: David Corfield on August 24, 2011 5:38 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

Are there any other kinds of arrow between universes?

Sure: logical functors come to mind from a topos-theoretic perspective, or the “inclusion” LVL\subseteq V and its relatives. (I put “inclusion” in quotes, because considered as a functor between models of ETCS, it’s faithful but not full.) Maybe the multiverse is more than just a 2-category? There’s a very natural proarrow equipment consisting of toposes, geometric morphisms, and lex functors, and I think all these other sorts of morphism are probably particular kinds of lex functors; I wonder whether they can be identified from the structure of that proarrow equipment?

E.g., in the category of groups one could say a property is necessarily held by a group if it is held at all codomains of arrows whose domain is that group.

Well, the category of groups has a zero object, so there is a morphism from any group to any other group; thus the only necessary closed sentences would be those that hold of all groups. That’s going to be true of many “algebraic” categories.

One thing to do would be to restrict the morphisms and consider only quotients, or only subgroups, and we do occasionally talk about modalities of this sort. For instance, a property is sometimes called “hereditary” if it is inherited by all subobjects.

Alternatively, we may get interesting things by thinking of properties of elements. For instance, I think an element of a ring is “possibly invertible” just when it lies outside of some maximal ideal. Maybe the modal logic of categories is worthy of further thought? It might be an interesting unifying principle.

Posted by: Mike Shulman on August 24, 2011 11:36 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

Alternatively, we may get interesting things by thinking of properties of elements.

I was wondering, during the last bout of modal logic thinking, how one might need to augment modal logic in the first order case if there are multiple arrows from one object to another. For an element of an object, you would want to express the difference between some property holding at images: for every arrow from that object; for every arrow to some object; for some arrow to every object; for some arrow to some object.

In the topological semantics approach, this plays on the possibility of multiple counterparts at another fibre of an element of a fibre, according to the path taken in the base space.

Posted by: David Corfield on August 25, 2011 8:59 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

I don’t really understand. I was thinking that if ϕ\phi is a property saying something about an object AA and possibly its elements, then ϕ\Box\phi means that (for any object BB and) for any morphism f:ABf\colon A\to B, the corresponding property to ϕ\phi (pushed forwards along ff) holds of BB. And that ϕ\diamondsuit\phi means that (for some object BB and) for some morphism f:ABf\colon A\to B, the corresponding property to ϕ\phi (pushed forwards along ff) holds of BB. How does that correspond to your English phrasings?

Posted by: Mike Shulman on August 27, 2011 2:37 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

In first-order modal logic, there is a notion of a collection of worlds related by an accessibility relation, and for each world a set of individuals, satisfying various properties. Then, according to David Lewis’s counterpart theory, counterfactual statements are made true by the state of affairs concerning counterparts in other possible worlds.

So, “Had I taken an aspirin, I wouldn’t have a headache now” is made true by events in the closest possible world to ours in which my counterpart took an aspirin, so long as there he has no headache.

Leaving aside the metaphysics of it all, there could be something useful in devising a model made up of a collection of worlds, related to each other in a certain way, and a collection of individuals at each world, such that I may be able to speak of xx in world 1 having a counterpart yy in world 2. Then we could wonder whether world 2 might contain more than one counterpart of xx. (This sort of property is considered in Principles that are not accepted in normal CT.)

Then we might consider a copresheaf of individuals over a category of worlds, and for xx at object AA, two counterparts of xx at object BB, according to different morphisms f,g:ABf, g: A \to B. Then the modal properties of xx are not restricted to ‘at all worlds all counterparts of xx…’, and ‘at some world some counterpart of xx…’. There are also ‘at all worlds some counterpart of xx…’, and ‘at some world all counterparts of xx…’.

Posted by: David Corfield on August 30, 2011 10:46 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

Can you give any non-metaphysical examples of where such modal properties might be useful? It seems more natural to me to quantify over objects and morphisms at the same time (that is, to quantify over objects in a slice category).

Posted by: Mike Shulman on September 1, 2011 5:05 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

If we’re just talking about objects and morphisms, might I not be interested in a space XX and a subspace YY, and ask of XX’s Moore path category whether it is the case that from a given object xx, for every object there is a morphism to it only meeting YY if at all at its endpoints? And I might wonder if for some object every morphism to it from xx avoids YY.

Seems like the flexibility our old friend dependent type theory gives us.

Posted by: David Corfield on September 1, 2011 5:39 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

Okay; I guess I see how you could call that a “modal property”. I don’t think I (yet) gain anything conceptually from thinking about it that way, though. With the ones I was calling \Box and \diamondsuit, I can see how the English modal words “necessarily” and “possibly” more or less fit the mathematical meanings.

Posted by: Mike Shulman on September 6, 2011 6:45 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

…making the multiverse into a 2-category.

So what kind of 2-category is it? A while ago we spoke of models of ETCS as forming the 2-category of well-pointed toposes with a natural numbers object satisfying the axiom of choice. Is replacement expressible naturally in category theoretic terms? Will there be a forgetful functor from models of ETCS + R to models of ETCS, and then a left adjoint?

Can one do a Tannakian trick and extract a theory of sets from its 2-category of models?

Posted by: David Corfield on August 24, 2011 9:54 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

There are category-theoretic ways to express replacement; one of these days I’ll get around to nLabifying them. The one I like best is in my paper on stack semantics.

Will there be a forgetful functor from models of ETCS + R to models of ETCS, and then a left adjoint?

I would say certainly yes, and probably not, respectively. ETCS+R is proof-theoretically stronger than ETCS, so if I just give you a model of ETCS, then you can’t construct from it any model of ETCS+R—unless either ETCS is inconsistent, or you make use of an ambient theory that’s at least as strong as ETCS+R. (As an example of the latter, in a sufficiently strong ambient theory you could consider a category of (pre)sheaves on your (small) model of ETCS, then do some sort of ultrapower to make yourself well-pointed again, but something like that seems to me unlikely to produce such an adjoint.)

Actually, from this perspective, I think it’s more natural to consider the 2-category of Boolean topoi with NNO and satisfying internal AC—i.e. topoi whose internal logic (or, more precisely, stack semantics) validates ETCS. (The stack-semantics version of replacement is called being “autological”.) When we move “into” a given universe, that means working in its internal logic, from which viewpoint it automatically “looks” well-pointed. But from the “external” viewpoint of the multiverse, there’s no need to demand anything be well-pointed, especially since most interesting constructions on topoi don’t preserve well-pointedness.

However, I think that modification is unlikely to change the situation regarding adjoints.

Can one do a Tannakian trick and extract a theory of sets from its 2-category of models?

My initial reaction is that all of these universes of sets are models of a single theory, so there is no reconstruction to be done. But maybe I misunderstand?

Posted by: Mike Shulman on August 24, 2011 7:03 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

I meant more the idea of reconstructing a theory from its collection of models, as in extracting the theory of groups from GrpGrp, or in Awodey and Forssell’s logical duality.

If I give you the 2-category (or whatever) of models of ETCS, can you reconstruct ETCS?

Posted by: David Corfield on August 25, 2011 8:48 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

I meant more the idea of reconstructing a theory from its collection of models

Yes, that’s what I thought you meant. What I meant was, there’s only one theory around (ETCS). But I guess we could consider it inside the class of first-order logical theories, and apply any general result to that. I don’t know exactly what sort of morphism between models of ETCS you would get falling out of that, though; I kind of suspect the resulting “category of models of ETCS” would not be any of the usual ones we like to think about. But I could be wrong.

Posted by: Mike Shulman on August 27, 2011 2:34 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

I can’t resist quoting one more lovely passage from the article:

In algebra, when one group is constructed from another or when a field is constructed as a quotient of a ring, these new structures are taken to be perfectly good groups and fields. There is no objection in such a case announcing “but that isn’t the real ++” analogous to what one might hear about a nonstandard or disfavored set-membership relation: “that isn’t the real \in.” Such remarks in set theory are especially curious given that the nature and even the existence of the intended absolute background notion is so murky.

Posted by: Mike Shulman on August 25, 2011 4:24 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

Well, I’ve got a naive question: doesn’t each axiomatic formulation of set theory correspond to a different (albeit “equivalent” in some sense) version of the topos Set?

Posted by: Alex on August 27, 2011 4:35 PM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

Any given model of any given (membership-based) set theory produces a category of sets. Different models of the same set theory may produce inequivalent categories, although the categories will share the same basic properties provable from the axioms of that set theory (like being a well-pointed topos, for instance).

Models of different membership-based set theories, on the other hand, may produce categories of sets that differ in their basic properties. For instance, whether or not the axiom of choice is assumed determines whether all epimorphisms in Set are necessarily split.

But on the third hand, models of two different set theories could produce equivalent categories of sets. For instance, the category of sets in a model of a non-well-founded set theory, and its subcategory of well-founded sets (which form a model of a well-founded set theory), will often be equivalent—for if we assume AC, then every set is bijective to a well-founded one (namely, a von Neumann ordinal).

On the fourth hand, for a large swath of membership-based set theories, the model of the original theory (assuming we fix a particular theory) can be recovered from the resulting category of sets, by constructing explicit membership trees; see the nLab entry pure set.

Does any of that help answer your question?

Posted by: Mike Shulman on August 29, 2011 6:02 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

It’s sometimes useful to have a popularised article as a starting point for further exploration. New Scientist is usually rather bad on math, but they did put Joel Hamkin’s idea nicely into context by quoting him in this article

Posted by: M. on September 5, 2011 7:50 AM | Permalink | Reply to this

Re: The Set-Theoretic Multiverse

The multiverse view, therefore, does not reduce via proof to a brand of formalism. In particular, we may prefer some of the universes in the multiverse to others, and there is no obligation to consider them all as somehow equal.

The phrasing suggests that formalists cannot prefer some formal systems or models thereof to others and must feel some obligation to consider them all as somehow equal. I don't know any formalists like that.

Posted by: Toby Bartels on December 31, 2011 9:40 AM | Permalink | Reply to this

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