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Johnstones classic topos theory book talks at some length in its introduction about how category theory/topos theory suggest that we view the 'universe' in which mathematics takes place as consisting of 'dynamic sets'.

This is motivated in part by the fact that any topos can formulate an internal logic using its objects and arrows that enables it to make 'quasi-set theoretical statements' about its objects -- for example, he points out that the truth object of a topos allows us to convert the comprehension axiom into an elementary statement about adjoint functors. Johnstone claims that this, together with many other nice facts about topoi, lead the budding toposopher to view an arbitrary topos as the correct place to formulate mathematics instead of a fixed universe of sets together with some axioms.

In his own words:

What, then, is the topos-theoretic outlook? Briefly, it consists in rejection of the idea that there is a fixed universe of "constant" sets within which mathematics can and should be developed, and the regognition that a notion of 'variable structure' may be more convieniently handled within a universe of 'continuously variable' sets than by the method, traditional since the rise of abstract set theory, of considering separately a domain of variation (i.e. a 'topological space') and a succession of constant structures attached to the points of its domain.

On the other hand, we have the set-theoretic multiverse philosophy put forward and explored by Hamkins which argues that the method of forcing in set theory suggests we view the background on which mathematics takes place as a multiverse of universes of sets, each with its own internal structural properties -- he also suggests here that the multiverse view de-emphasizes structural properties of individual universes, as opposed to a universe-based view which might suggest that we explore properties of individual highly structured universes.

In his own words:

The multiverse view in set theory is a philosophical position offered in contrast to the Universe view, an orthodox position, which asserts that there is a unique background set-theoretic context or universe in which all our mathematical activity takes place. (...) A paradox for the universe view, which I mention in the slides to which you link, is that the most powerful set-theoretic tools that have informed a half-century of research in set theory are most naturally understood as methods of constructing alternative set-theoretic universes. (...) The multiverse view takes these diverse models seriously, holding that there are diverse incompatible concepts of set, each giving rise to a set-theoretic universe in which they are instantiated. The set-theoretic tools provide a means of modifying any given concept of set to a closely related concept of set, whose resulting universes can be fruitfully compared in a single mathematical context.

I am more familiar with topoi and two-dimensional/inner category theory than with forcing and related multiversal notions, but these two views seem quite similar in that they suggest many different possible places for 'ordinary' mathematics to be formulated, with consequences for each choice of location. My question is:

What is the relationship between a toposophers view of mathematics and a set-theoretical multiverse philosophers view? Has there been any work done on the relationship between the two?

For example, could each possible universe be cast as a topos with some additional structure dependent on the axioms underpinning the universe or vice verse, at least up to a canonical isomorphism or equivalence?

Here I transcribe some relevant comments made below:

Noah Schweber suggests and Sridhar Ramesh confirms that if you take a model of ZF and construct a topos from it, you can recover the ZF model from the topos, by first recovering the ordinals (as the structures the topos considers to be well-ordered), and then carrying out the construction of the cumulative hierarchy relative to those ordinals and the toposes' notion of power objects.

Sridhar also points out that all the forcing constructions of set theory have very direct topos theoretic analogues. Forcing over a partial order in the traditional sense is the same as constructing the presheaf topos over that partial order and then passing through the double negation transformation to recover a Boolean topos rather than an intuitionistic one, as well as the fact that toposes are agnostic to distinctions like between ZF vs. ZF - Foundation + Aczel's Antifoundation Axiom, or the like. You can just as well recover from a ZF topos a model of ZF - Foundation + Aczel's Antifoundation Axiom, if you like, by constructing (essentially) the terminal coalgebra for the powerset functor instead of the initial algebra, or such things. Toposes only care about sets up to cardinality, and do not care about any canonical inner structure imposed upon them (toposes are "structural set theory", blind to "material set theory" distinctions).

It is also pointed out that you can analogously construct a model of ZF-Foundation+AAA from a model of ZF, and so on, but that given a model of some arbitrary material set theory, not presumed off the bat to be specifically a model of ZF or a model of Antifoundational ZF, you can tell which one the model is actually of (even though you can extract a corresponding model of the other in either case). Whereas with a topos generated from these, there's no notion of whether it's a Foundation topos or an Antifoundation topos; it's the same topos either way.

Alec Rhea
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  • Re: your last paragraph, doesn't each model $M$ of ZFC (or indeed vastly less) yield a topos with objects = elements of $M$ and morphisms = functions in $M$? – Noah Schweber Dec 18 '18 at 19:19
  • @NoahSchweber Yes, but it isn't clear (to me) that models of different collections of axioms would naturally give rise to 'different' topoi constructed as you suggest -- is this always the case? It seems (very naively) like we could construct an isomorphism between the toposes of any two models (as you've defined them) of the same cardinality. – Alec Rhea Dec 18 '18 at 19:23
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    I don't see why that's plausible - how would you, for example, try to construct an isomorphism between a well-founded model and an ill-founded model? Or a model in which CH holds versus a model in which CH fails? – Noah Schweber Dec 18 '18 at 19:27
  • @NoahSchweber I am not suggesting an isomorphism between the models, but one between the toposes you suggest building out of the models. In particular, the topos will (I believe) only pay attention to the notion of products and function spaces (cartesian closedness), the notion of subsets (subobject classifier), and something corresponding to the existence of finite limits, but I don't think it will be sensitive to finer properties of the models as you suggest. – Alec Rhea Dec 18 '18 at 19:30
  • Sorry, I meant to say "topos corresponding to that model." I still don't see how you propose to construct such an isomorphism. In particular, it seems we can distinguish between true $\omega$ and an illfounded $\omega$: true finiteness is captured at the level of isomorphisms, and only true $\omega$ satisfies "every subobject of me is isomorphic to me or truly finite" (even if that can't be "expressed" appropriately internally). – Noah Schweber Dec 18 '18 at 19:35
  • @NoahSchweber That is interesting, but I am not well-versed enough in ill-founded model theory to know if there might be some other object we can send 'true' $\omega$ to in the topos of the ill-founded model. It seems like some ill founded models still have well founded ordinals (https://mathoverflow.net/questions/227418/ill-founded-models-of-set-theory-with-well-founded-ordinals), but I take it from your suggestion that there are some ill-founded models where even $\omega$ is not well-founded. For these cases, you may be right that the topos you suggest is nonisomorphic to the one (cont.) – Alec Rhea Dec 18 '18 at 19:40
  • built from it's ill founded model, but this still leaves open the more general case of when two models of differing axiom collections will give rise to different topoi (as you suggest building them). – Alec Rhea Dec 18 '18 at 19:41
  • Yes, you can easily make the ordinals (including $\omega$) ill-founded: just use compactness. (That question is about models of set theory without foundation, where the ill-foundedness can occur "off to the side.") – Noah Schweber Dec 18 '18 at 19:53
  • Actually it seems to me that the theory of the topos (viewed itself as a first-order structure) lets you recover the theory of the original model, since we can talk about elements (= morphisms from the terminal object, modulo the appropriate commutativity condition). – Noah Schweber Dec 18 '18 at 19:59
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    Yes, if you take a model of ZF and construct a topos from it, you can recover the ZF model from the topos, by first recovering the ordinals (as the structures the topos considers to be well-ordered), and then carrying out the construction of the cumulative hierarchy relative to those ordinals and the toposes' notion of power objects. – Sridhar Ramesh Dec 18 '18 at 20:15
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    It should also be noted that all the forcing constructions of set theory have very direct topos theoretic analogues. Forcing over a partial order in the traditional sense is the same as constructing the presheaf topos over that partial order and then passing through the double negation transformation to recover a Boolean topos rather than an intuitionistic one. – Sridhar Ramesh Dec 18 '18 at 20:22
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    It's worth noting, though, that toposes are agnostic to distinctions like between ZF vs. ZF - Foundation + Aczel's Antifoundation Axiom, or the like.

    You can just as well recover from a ZF topos a model of ZF - Foundation + Aczel's Antifoundation Axiom, if you like, by constructing (essentially) the terminal coalgebra for the powerset functor instead of the initial algebra, or such things.

    Toposes only care about sets up to cardinality, and do not care about any canonical inner structure imposed upon them (toposes are "structural set theory", blind to "material set theory" distinctions).

     – Sridhar Ramesh Dec 18 '18 at 20:23
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    @SridharRamesh Re: your last comment, I don't think that's really different from the ZF setting: you can analogously construct a model of ZF-Foundation+AAA from a model of ZF, and so on. (Also, my previous comment had a silly error: you can define "terminal object-ness," and you need the commutativity condition to distinguish two maps from different terminal objects. But the point stands.) – Noah Schweber Dec 18 '18 at 20:25
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    Right, it's not a difference except in that, given a model of some arbitrary material set theory, not presumed off the bat to be specifically a model of ZF or a model of Antifoundational ZF (but, let's say for now, presumed to be one of these two), you can tell which one the model is actually of (even though you can extract a corresponding model of the other in either case; the two are different but related models). Whereas with a topos generated from these, there's no notion of whether it's a Foundation topos or an Antifoundation topos; it's the same topos either way. – Sridhar Ramesh Dec 18 '18 at 20:29
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    Or, illustrated perhaps more simply, given a model of material set theory, not presumed off the bat to be a model of ZF, one can ask questions like "Are there any sets in here which contain themselves?". Two models of material set theory may differ in their answers to questions like these. Whereas such questions cannot be asked of a topos; when generating a topos from a model of material set theory, it loses information about such distinctions. – Sridhar Ramesh Dec 18 '18 at 20:35
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    @SridharRamesh Yes, that's a good point. – Noah Schweber Dec 18 '18 at 20:39
  • @SridharRamesh and NoahSchweber, thank you both for your comments; I have added them to the main body of the post as I find them fascinating and relevant. – Alec Rhea Dec 18 '18 at 20:42
  • Which foundations of mathematics are you assuming? (I think this is a necessary question if we want to compare the two "philosophies", and even before that, if we want to understand what a "comparison" could actually mean...) – Qfwfq Dec 18 '18 at 20:42
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    For a technical acount of how to go back and forth between toposes and model of ZF I recommand reading MIke shulman's paper "https://arxiv.org/abs/1004.3802". The short answer is that these two point of view are indeed very similar up to small techincal difference: the difference between mateial/structural set theory mentioned in previous comment, toposes allows for intuitionistic logic, and toposes comes without the unbounded replacement axiom (which make then quite weaker than ZF in terms of proof theoretic strength) – Simon Henry Dec 18 '18 at 21:03
  • @SimonHenry Re: the second point, we can of course look at intuitionistic ZF, so that's not an insurmountable difference. – Noah Schweber Dec 18 '18 at 21:19
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    We can look at intuitionistic ZF - Infinity - Replacement and with Separation restricted to bounded formulas and with possible urelements, if we want to take the difference completely to zero. Any such theory gives rise to an elementary topos, and vice versa, in such a way that the concept of an elementary topos could be said to amount to the same concept as a structural set theory extending this base theory. Adding back in Infinity amounts to demanding the topos have a natural numbers object; one can also demand the topos be Boolean, add in the Axiom of Choice, etc, if one likes. – Sridhar Ramesh Dec 19 '18 at 02:12
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    I've phrased this in terms of extensions of theories, instead of in terms of models, because the former is really the most natural way to look at all this. But classical models are then those particular Boolean theories which are complete and consistent and have existential witnesses; i.e., Boolean toposes on which the Hom(1, -) functor preserves coproducts and epics (equivalently, preserves coproducts and is faithful). – Sridhar Ramesh Dec 19 '18 at 02:16
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    @SimonHenry might as well recommend the new version of the first half of that paper: https://arxiv.org/abs/1808.05204 (I don't know when the second half will come out as a separate article, but that's not what's relevant here). – David Roberts Dec 19 '18 at 05:14
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    @SridharRamesh I’m always confused when you category guys seem to blur the distinction between a mathematical structure and the theory it satisfies. What’s going on with that? – Monroe Eskew Dec 19 '18 at 07:34
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    A structure is just a special case of a theory: the case where provability commutes with every logical operator (i.e., where "A v B" is provable just in case "A" is provable or "B" is provable; "∃ x. P(x)" is provable just in case there exists a term t such that P(t) is provable, etc.).

    Often, results that are traditionally phrased in terms of models are really about theories in general, and gain a greater clarity and unifying perspective by being phrased as such (plus, constructivity from not appealing to ultrafilters needlessly). This is also the start of Boolean-valued models, etc.

     – Sridhar Ramesh Dec 19 '18 at 18:37
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    The nice thing about theories is that they are "essentially algebraic" structures and thus have all the usual algebraic properties; there are free constructions, products/coproducts/limits/colimits of arbitrary theories, and the like available, which generally isn't true for models of non-algebraic logics (unless you do things like allow models valued in arbitrary Boolean algebras, blurring the theory/model distinction again). – Sridhar Ramesh Dec 19 '18 at 18:45
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    @SridharRamesh- What informs this use of terminology? It’s against the mainstream usage in logic. A theory is a collection of sentences in a language. A structure is a collection of things with functions and relations. The satisfaction of sentences by a structure is defined and we ask whether the structure satisfies a theory. Sometimes such questions are hard. There are often many nonisomorphic structures satisfying a given theory, and often many interpretations of the language in a structure (this is relevant in, for example, rigidity). These are just not the same kind of thing. – Monroe Eskew Dec 19 '18 at 20:37
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    Furthermore, provability commutes with logical operators only for complete theories, which are usually not computable. And complete theories cannot be identified with structures because there are often many structures satisfying complete theories, many interpretations of a theory in a structure, etc. – Monroe Eskew Dec 19 '18 at 20:42
  • Details on how forcing is a topos-theoretic construction are referenced here: https://ncatlab.org/nlab/show/forcing#ReferencesInTermsOfClassifyingToposes – Urs Schreiber Dec 20 '18 at 10:47
  • @MonroeEskew: They're exactly the same thing. Given a theory of the sort I described, there is a corresponding structure, and vice versa. The two are precisely the same data. This is the correspondence which associates to every structure its "elementary diagram" or "complete diagram", as the model theorists say. – Sridhar Ramesh Dec 20 '18 at 18:55
  • Yes, the particular kinds of theories which correspond to fully-specified structures (aka "models") are always complete and oft uncomputable; that is the nature of the properties satisfied by a model. If you are unhappy with that, you should be happier with considering (possibly incomplete, etc) theories in general and not only fully-pinned-down models in particular, just as I was advocating. You can now understand where I am coming from, in preferring to speak in terms of extensions of theories vs. phrasing everything via full-on models, the former avoiding nonconstructive ultrafilters, etc. – Sridhar Ramesh Dec 20 '18 at 19:05
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    @SridharRamesh No- other nonisomorphic structures B may satisfy the elementary diagram of A. Or we may be able to assign the new constants differently to get an nontrivial elementary embedding of A into itself. It’s not the same notion. – Monroe Eskew Dec 20 '18 at 19:07
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    I never denied that other nonisomorphic structures B may satisfy the elementary diagram of A (just as one theory may extend another). I never claimed A to be the only structure satisfying A's elem. diag. I am just saying, the data of A and of A's elem. diag. are the same. There is a one-to-one correspondence between "structures" and "theories which are elementary diagrams", and so via this correspondence, I think of structures as special cases of theories. And in this way, many things others view in terms of models of theories, I view in terms of theories extending other theories in general. – Sridhar Ramesh Dec 20 '18 at 19:15
  • Anyway, the whole reason I brought this up in this context is that people keep speaking about extracting models of ZF from toposes, but it's worth noting that a topos in general (even a Boolean topos) does not give rise to a model in the traditional sense, but rather corresponds to a general theory which may or may not be complete, because the Heyting algebra of subobjects of 1 in a topos may comprise something other than just the two values True and False. – Sridhar Ramesh Dec 20 '18 at 19:18
  • The most natural way to look at things is that a topos does not correspond to a model of set theory, as such, but rather, a topos corresponds to a theory. A topos corresponds to a structural set theory in intuitionistic logic (aka, a theory in higher-order intuitionistic logic). There is a one-to-one correspondence again; from any such theory, you get a topos, and vice versa in an inverse way. – Sridhar Ramesh Dec 20 '18 at 19:20
  • Specifically, the objects of the topos are the definable sets of the theory, the morphisms of the topos are the definable functions of the theory, and the equations between compositions in the topos (its 2-cells) are the provable such equations of the theory. – Sridhar Ramesh Dec 20 '18 at 19:23
  • Further, the study of models of PA shows that it can be a different question whether the model has a computable presentation versus whether its theory is computable. – Monroe Eskew Dec 20 '18 at 19:24
  • A topos corresponds to a structural set theory in intuitionistic logic in precisely the same way that a category with finite products corresponds to a multisorted algebraic theory. This is my main point. A general topos does not directly correspond to a "model" of a set theory. A general topos corresponds to a theory, which may or may not be the elementary diagram of a model. Everything about the relationship between toposes and ZF, etc, becomes clearest from this perspective. – Sridhar Ramesh Dec 20 '18 at 19:29
  • The topos most directly relevant to ZF (or whatever set theory you are interested in studying, but for now let's say ZF) is the topos whose objects are the definable sets in ZF, whose morphisms are the definable functions in ZF, and whose equalities between compositions of morphisms are the provable such equations in ZF. – Sridhar Ramesh Dec 20 '18 at 19:40
  • This is a rather "small" topos, in that it has only countably many objects and morphisms and 2-cells, but that's the nature of it; ZF is a countable theory and is encapsulated completely by this countable topos. This is also a topos where there are many more than two subobjects of 1, and again, this is to be expected and captures an important fact about ZF, that there are many more than two equivalence classes of sentences of ZF under ZF-provable bi-implication. – Sridhar Ramesh Dec 20 '18 at 19:41
  • (God, the character count restrictions... Why am I trying to carry out a mathematical discussion on a platform antipathetic to mathematical discussion?

    More important question, why have we built out our modern global framework for mathematicians to ask other mathematicians about mathematics in this way, with its major hubs explicitly designed to be antipathetic to discussion? [I mean, I know some of the reasons, but I'm still going to grumble...])

     – Sridhar Ramesh Dec 20 '18 at 19:50
  • Perhaps it makes more sense to identify a structure with its atomic diagram rather than elementary diagram. The latter has a lot more information. – Monroe Eskew Dec 20 '18 at 19:51
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    Sure, that's something I often do too (more often, indeed). What's relevant is whatever is relevant to the kind of logic one is working with and the kind of homomorphisms one wants to consider. If one just wants to consider algebraic properties and ordinary algebraic homomorphisms, the atomic diagram (sentences in just the language of algebra; equations between compositions of functions at various arguments) is the way to go. If one wants to consider disjunctive properties and homomorphisms which must preserve these disjunctive properties, add in the sentences characterizing these. And so on. – Sridhar Ramesh Dec 20 '18 at 20:05
  • A topos being a theory in higher-order logic, though, it corresponds directly to a diagram even more informative than a mere elementary (i.e., first-order) diagram. The kinds of toposes which correspond directly to models of ZF amount to an enumeration of all the sentences in the language of higher-order logic with parameters from and validated by these models.

    But, as noted, most toposes do not correspond directly to a model, but rather correspond more generally to a theory (possibly incomplete, possibly lacking existential witnesses, etc).

     – Sridhar Ramesh Dec 20 '18 at 20:15
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    @SridharRamesh and MonroeEskew, thank you both for having this discussion here -- I've found it very informative, and I think it's good to have discussions between experts available on a public forum. – Alec Rhea Dec 20 '18 at 21:14
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    @SridharRamesh- Something seems to be wrong here. The collection of definable sets in ZF is not necessarily a model of ZF. So I don’t know what you mean by “the topos of definable sets” “completely encapsulates” ZF. See here: https://mathoverflow.net/questions/10413/definable-collections-without-definable-members-in-zf – Monroe Eskew Dec 21 '18 at 09:45
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    @SridharRamesh I'm really confused about your last dozen comments. The category you describe is not a topos in general. Are you thinking of classifying toposes? That wouldn't work for ZF since it's not a geometric theory. – François G. Dorais Dec 22 '18 at 10:03
  • I do not mean "definable set" within a model, but to mean a set definable within the theory ZF itself. So let me be clear about the topos I was describing (this will take several comments because of character count restrictions): – Sridhar Ramesh Dec 23 '18 at 17:45
  • There is an elementary topos described as follows:

    The objects are formulas with one free variable $\phi(x)$ in the language of ZF such that ZF proves "There exists a unique set $S$ such that $\phi(S)$".

     – Sridhar Ramesh Dec 23 '18 at 17:47
  • The morphisms (up to an equivalence relation to be described next) between objects $\phi(x)$ and $\psi(x)$ are formulas with two free variables $R(a,b)$ in the language of ZF such that ZF proves "Letting $S$ be the unique set such that $\phi(S)$ and $T$ be the unique set such that $\psi(T)$, we have that for every $a \in S$, there is a unique $b \in T$ such that $R(a, b)$. – Sridhar Ramesh Dec 23 '18 at 17:47
  • And two such morphisms $R(a, b)$ and $S(a, b)$ are considered equal just in case ZF proves "Letting $S$ be the unique set such that $\phi(S)$ and $T$ be the unique set such that $\psi(T)$, we have that for every $a \in S$, there is a $b \in T$ such that $R(a, b)$ and $S(a, b)$". – Sridhar Ramesh Dec 23 '18 at 17:47
  • Composition of morphisms $R(a, b) : \phi(x) \to \psi(x)$ and $S(a, b) : \psi(x) \to \gamma(x)$ is given in the expected way for composing binary relations, by the formula $(S \circ R)(a, b) =$ "$\exists m \in T$ such that $R(a, m) \wedge S(m, b)$ (where $T$ is the unique set such that $\psi(T)$)". – Sridhar Ramesh Dec 23 '18 at 17:54
  • This is the elementary topos that I consider to directly represent the theory ZF. Is there controversy as to my claim that this is indeed an elementary topos? – Sridhar Ramesh Dec 23 '18 at 17:56
  • @SridharRamesh You may define this structure among formulas, but what is the claim about ZF? Just that you can define the theory ZF given the topos? – Monroe Eskew Dec 23 '18 at 19:49
  • The claim is that this is the topos which most directly captures ZF. That's a human claim, not a math claim, but if you want a math claim, functors from this topos to Set which preserve all elementary topos structure correspond to models of ZF whose internal powersets are indeed full powersets, functors from this topos to Set which preserve first-order structure but not necessarily power objects correspond to models of ZF as a first-order theory in general, and all the forcing constructions can be carried out relative to this topos to prove independence of statements from ZF. – Sridhar Ramesh Dec 23 '18 at 20:08
  • Furthermore, functors from(/to) this topos to(/from) other toposes preserving appropriate structure correspond to interpretations of ZF into(/into ZF of) other structural set theories, and the analogous construction can be carried out and the analogous results hold when ZF is replaced by any other set theory (which internally proves Set to be a topos). – Sridhar Ramesh Dec 23 '18 at 20:12
  • The simpler claim is that this structure is indeed an elementary topos. I note this simpler claim only because there seemed to be some confusion earlier, where Francois Dorais felt the structure I was describing was not an elementary topos. I take it this was perhaps because of some ambiguity in phrases like "ZF definable set", and that now that I have clarified the category I have in mind, all will agree this structure is indeed a well-defined elementary topos? – Sridhar Ramesh Dec 23 '18 at 20:22
  • (Actually, for handling the nuances of how ZF allows unbounded separation and replacement, we really should be looking at the non-topos category constructed in the same way except for that its objects correspond to definable classes and not just sets (a full subcategory of which will be the elementary topos of just the sets), but I trust it's ok if I sweep that under the rug for now...) – Sridhar Ramesh Dec 23 '18 at 20:30
  • I don’t see the point of restricting to formulas which provably define a unique set, nor do I see why this should allow enough objects. There are formulas which probably define a unique set, for which ZF cannot decide whether the set defined is empty. – Monroe Eskew Dec 23 '18 at 20:51
  • "Nor do I see why this should allow enough objects": Enough objects for what purpose? The only purpose of this topos is to faithfully capture ZF. – Sridhar Ramesh Dec 23 '18 at 20:54
  • "There are formulas which probably define a unique set, for which ZF cannot decide whether the set defined is empty". Yes, this is equivalent to the fact that ZF is not a complete theory. That's fine. That's an important fact about ZF. And in the resulting topos, it is reflected in the existence of objects which are neither initial, nor is their map to the terminal object epic. Equivalently, the subojects of the terminal object are not only 0 and 1. That's fine. Most toposes are like that. Again, it just means being an incomplete theory, an important fact about ZF to capture. – Sridhar Ramesh Dec 23 '18 at 20:56
  • Oh, in listing mathematical claims above, I forgot to state the most important mathematical claim about this topos: a sentence is true in its "internal language" just in case that sentence is provable in ZF. That's what makes this topos the most direct representation of ZF. Toposes generated from specific models of ZF don't have this property. – Sridhar Ramesh Dec 23 '18 at 21:00
  • Why is it the case that when you take only the formulas in one free variable such that ZF proves there is a unique object satisfying the formula, then you get the full set of consequences of ZF as the things true in the internal logic? What is that definable-sets thing doing for you? – Monroe Eskew Dec 23 '18 at 21:06
  • It's giving you the objects of a topos, that's all. That's not the important bit, you can just as well make a category where objects correspond to arbitrary unary formulas in ZF, now thought of as representing the class (possibly a proper class) of all sets satisfying that property. Morphisms, etc, can be defined as before, mutatis mutandis. This gives us a nice category too; just not a topos (those objects coming from formulas whose extension is not ZF-provably a set will not have power objects). The previously described topos will be a full subcategory of this. – Sridhar Ramesh Dec 23 '18 at 21:15
  • If you like, imagine that instead of saying "unary formulas such that ZF proves a unique set satisfies the formula", I had instead said "unary formulas such that ZF proves the collection of things satisfying the formula comprises a set (as opposed to a proper class)", and changed everything else accordingly. It's a minor change, no big deal, it doesn't change the resulting category, but maybe it's more to your taste to phrase it this way. – Sridhar Ramesh Dec 23 '18 at 21:19
  • So you’re just constructing some objects because you want to have objects. But your main goal is to have things true in the internal logic to be exactly the closure of ZF under logical consequence. (Can you say why this is the case?) And functors to Set that preserve such-and-such structure yield classical models of ZF. Sorry if this sounds offensive, but what is the payoff for classical mathematics? – Monroe Eskew Dec 23 '18 at 21:25
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    The discussion in this question and thread had been about the correspondence between set theories and toposes, and all I'm doing is noting how that correspondence works. Working within ZF is the same as working internally to the topos I have just described. If you are doing mathematics within ZF, and you want to know what topos that corresponds to, this is the topos it corresponds to. To demonstrate that statements are unprovable in ZF, take this topos as your base topos and then carry out the suitable sheaf constructions. Etc. I don't know what other kind of "payoff" you're looking for. – Sridhar Ramesh Dec 23 '18 at 21:31
  • "But your main goal is to have things true in the internal logic to be exactly the closure of ZF under logical consequence. (Can you say why this is the case?)": Well, for example, if some logician wants to demonstrate, in a topos-theoretic way, that the provability of A in ZF entails the provability of B in ZF (as in the independence results), it is very useful to have a topos which validates just those propositions which are provable in ZF! Then you carry out whatever forcing or other construction relative to this ground topos, to demonstrate the desired result. – Sridhar Ramesh Dec 23 '18 at 21:47
  • When I asked, “Can you say why this is the case?” I was asking why does the category constructed with these particular objects (you gave 3 different choices) result in a topos where truth = ZF-provable? – Monroe Eskew Dec 24 '18 at 08:31
  • If I’m getting the gist of it, I think you could alternatively take the objects to be “formulae in one free variable which ZF proves to be satisfied on a finite set of integers.” Or take your objects to be sentences in the language of ZF with provable implications as the arrows. – Monroe Eskew Dec 24 '18 at 09:10
  • It you take your objects to be sentences and provable implications as arrows (presumably identifying all parallel arrows?), you don't get a topos. You just get a partial order (a Boolean algebra, specifically; the Lindenbaum-Tarski algebra, as people call it). That's fine, it's an object of interest too, but it doesn't give you something you can use as a base topos for the forcing, etc, constructions. In the topos I am concerned with, this algebra appears as the full subcategory of subterminal objects, but there is much more structure present as well in the other, non-subterminal objects. – Sridhar Ramesh Dec 24 '18 at 18:28
  • Similarly, restricting to just the formulas whose extensions are provably finite gives you the full subcategory of my topos consisting of finite sets. That's a topos in itself, but not the relevant one if your goal is to study ZF, as such. It's a context that considers all sets to be finite, which is very different from ZF's position on the matter. – Sridhar Ramesh Dec 24 '18 at 18:30
  • For the record, the topos I've been describing is a full subcategory of the category sometimes called "the syntactic category of ZF". The syntactic category of ZF has all the definable classes of ZF as objects. It's a Boolean pretopos but not a topos, as only those classes which are provably small (aka sets) will have power objects. By restricting to those, we obtain the relevant topos. The full syntactic category is really what's of interest, though. It's an instance of the structure sometimes called a "category of classes", meaning just the appropriate augmentation of the concept of a topos. – Sridhar Ramesh Dec 24 '18 at 19:02
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    Please explain why truth = provability in ZF In one of the toposes you defined. I already pointed out some problem with the definable-sets category, and Dorais seemed to agree there was something wrong, and then you said the objects could be chosen differently. So I’m guessing the idea is some kind of syntactic trick going on to achieve the truth values you want (having to do with the fact that the objects are formulas?), which seemed to be the main point. What’s the argument? – Monroe Eskew Dec 25 '18 at 06:31
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    What you’re saying deviates from the ncatlab page on syntactic categories in several ways. For one, provable definability doesn’t come up there. So it’s hard for the nonexpert to follow you. – Monroe Eskew Dec 25 '18 at 06:56

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