Timeline for Large categories vs. $\mathrm{U}$-categories: why is the loss of category-theoretic information inessential?
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2018 at 18:11 | vote | accept | Jxt921 | ||
| Sep 5, 2017 at 1:10 | comment | added | Joel David Hamkins | Oh, sorry, I don't use chat. By that remark, I meant that we augment the language of set theory with an additional unary predicate symbol $I$. It is still a one-sorted theory, since $I$ is a predicate, not an object, but by this means $I$ becomes a definable class in the expanded language. We will have $ZFC(I)$, which includes the replacement axiom in the language with I, but the axiom scheme we were discussing includes instances of elementarity for $V_\gamma$ and $V_\delta$ for $\gamma,\delta\in I$ only with respect to assertions in the ordinary language of set theory. | |
| Sep 5, 2017 at 1:06 | comment | added | Jxt921 | @JoelDavidHamkins I've tried to write to you in chat, but I'm not sure if this worked; I've never did anything with chats on stackexchange. What I've tried to ask you in chat is this: "could you please elaborate on what you mean by the "language of set theory augmented with a symbol for $I$"? It means that we have a two-sorted (on contrast with pure ZFC, which is one-sorted) first-order theory with sets in $V$ (which itself doesn't exists in ZFC) and a class $I$?" | |
| Sep 2, 2017 at 23:56 | comment | added | Jxt921 | Let us continue this discussion in chat. | |
| Sep 2, 2017 at 15:31 | comment | added | Joel David Hamkins | Regarding your inquiry about how to define that $\phi$ holds in $V_\lambda$, you have the right idea: one needs only to bound the quantifiers of $\phi$ by $V_\lambda$. More generally, ZFC is able to define the notion of truth in any set-sized structure in any language. So in ZFC, we can refer to any statement $\phi$ being true or not in any structure for that language. | |
| Sep 2, 2017 at 15:30 | comment | added | Joel David Hamkins | So it is inherent in the elementary-chain idea that one is using only some of the inaccessible cardinals, as otherwise one will definitely not have an elementary chain. | |
| Sep 2, 2017 at 15:29 | comment | added | Joel David Hamkins | I'm glad you found my answer helpful, and thanks for accepting and for the bounty. The axiom I stated is expressible as a scheme in the language of set theory augmented with a symbol for I. Your stronger assertion about all inaccessible won't work, since it is never true that one can get the class of all inaccessible cardinals to form an elementary chain. After all, if $\kappa<\gamma<\delta$ are successive inaccessible cardinals, then $V_\gamma$ will think "$\kappa$ is the largest inaccessible cardinal" but $V_\delta$ will not agree. | |
| Sep 2, 2017 at 15:11 | comment | added | Jxt921 | What is more, how to we define in the language of ZFC that $\phi(a_1,...,a_n)$ "holds" in $V_\lambda$ and in $V_\mu$? We need to "restrict" possible quantifiers to $V_\lambda$ and $V_\mu$, that is, if there is a term $\forall x, P(x)$ in $\phi$, then we need to change it to $\forall x, x \in V_\lambda \to P(x)$ and to $\forall x, x \in V_\mu \to P(x)$, respectively, and we need to do the same with $P(x)$. | |
| Sep 2, 2017 at 15:08 | comment | added | Jxt921 | if and only if it holds in $V_\mu$. However, it seems to be stronger than the one axiom we need, as it implies not the existence of some class of inaccessibles such that for $\lambda < \mu, V_\lambda$ is elementary with respect to $V_\mu$, but rather that for all inaccessibles $\lambda < \mu$ implies that $V_\lambda$ is elementary with respect to $V_\mu$. | |
| Sep 2, 2017 at 15:06 | comment | added | Jxt921 | @JoelDavidHamkins Dear Joel, your answer is really helpful. However, what I wonder how we can state this stronger axiom of universes (or the axiom of elementary universes, or whatever we can call this) purely in the language of ZFC. One way was to separate it into the axiom of universes ($\exists$ inaccessilbe $\lambda$, $\forall$ inaccessible $\lambda \ \exists$ inaccessible $\mu$ such that $\kappa < \mu$) and the axiom schema that given a formula $\phi(x_1,....,x_n)$, if $\lambda < \mu$ are two inaccessibles, then $\forall a_1,...,a_n \in V_\lambda, \phi(a_1,...,a_n)$ holds in $V_\lambda$ | |
| Sep 2, 2017 at 10:50 | history | bounty ended | Jxt921 | ||
| Aug 28, 2017 at 16:10 | comment | added | Sridhar Ramesh | Ah, of course; silly of me not to see that. Thanks for clarifying! | |
| Aug 28, 2017 at 12:35 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
fixed typo
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| Aug 28, 2017 at 11:50 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Fixed subtle issue with nonstandard formulas
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| Aug 28, 2017 at 11:21 | comment | added | Joel David Hamkins | @SridharRamesh The fact that the union of an elementary chain is still elementary is a basic theorem of model theory. So indeed, each $V_\delta$ is elementary in $V$, and it doesn't matter which elementary chain you use. David, one gets $\bigcup_\delta V_\delta=V$ for any unbounded class of cardinals, since the $V_\delta$ are linearly ordered by inclusion. | |
| Aug 28, 2017 at 11:18 | comment | added | Joel David Hamkins | @MikeShulman Yes, if you drop the inaccessibility requirement, then this basically amounts to the Feferman theory. The existence of an elementary chain of rank-initial segments $V_\gamma\prec V_\delta\prec\cdots$ unbounded in the universe is equivalent over ZFC to the existence of a truth predicate for first-order truth, which is much weaker than Mahlo (or even just one inaccessible), but still stronger than ZFC. If one only wants $V_\delta\prec V$ as a scheme for each $\delta\in I$, then it is equiconsistent with ZFC. | |
| Aug 28, 2017 at 10:45 | comment | added | Mike Shulman | Also, ZMC/S is a slight modification of Feferman's propose theory ZFC/S, which assumes only that S is an elementary substructure of the universe (rather than a Grothendieck universe; so in particular it only satisfies "first-order replacement" rather than "second-order replacement"). I presume one could instead assume an elementary chain of such substructures to obtain a version of your answer that doesn't extend the consistency strength of ZFC (though, as pointed out in my notes, the lack of second-order replacement is annoying for category theory). | |
| Aug 28, 2017 at 10:42 | comment | added | Mike Shulman | This is an excellent answer. I just want to point out that it is very closely related to the theory ZMC/S proposed in my notes and mentioned in the P.S. of the OP; the latter basically amounts to fixing a particular symbol "S" to denote an arbitrary one of the universes in the elementary chain. | |
| Aug 28, 2017 at 3:24 | comment | added | David Roberts♦ | Wouldn't one want the condition $\bigcup_\delta V_\delta = V$ in order to conclude truth in $V$ from truth in some universe? Or is this obvious from the definition? | |
| Aug 28, 2017 at 1:38 | comment | added | Sridhar Ramesh | Why does the elementary chain universe axiom allow us to move from "Statement S is true in every Grothendieck universe, and thus in each Grothendieck universe in the proper class I" to "Statement S is true in the ambient set-theoretic universe"? Indeed, could there not be two proper classes I and I' of chains of elementary substructure Grothendieck universes such that members of I and members of I' are not elementarily equivalent? Perhaps there could not, but it's not obvious to me why not. | |
| Aug 27, 2017 at 22:34 | comment | added | David Roberts♦ | Sorry, I meant "a lot of constructions of interest, in order to be preserved, turn out to not to not need such elementary embeddings" :-) | |
| Aug 27, 2017 at 14:12 | comment | added | Joel David Hamkins | Thanks! I am not preserving elementary embeddings here, but rather, it is about the different universes agreeing with one another on the truth of mathematical assertions, which is the same thing that Low is doing for the restricted assertions he considers. My point here is that the method of my answer essentially continues Low's project by allowing you to go all the way, achieving full agreement on the truth of all assertions between any universes in the class arising from $I$. | |
| Aug 27, 2017 at 13:52 | comment | added | David Roberts♦ | The relationship to ORD is Mahlo is nice! In practice a lot of constructions of interest turn out to not need such elementary embeddings to be preserved, as in Low's paper. Unfortunately he only started the process of a full treatment. | |
| Aug 27, 2017 at 13:42 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |