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In this post, Joel David Hamkins (Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2, URL (version: 2015-05-10): https://mathoverflow.net/q/206178) answers a questions about categorical theories extending second-order ZFC.

The way he obtains the categoricity of a theory T is by adding to ZFC2 axioms that make reference to models of T, i.e., ZFC2 +"there is a unique model of T, and no inaccessible cardinals above the size of that model", thus relying on Zermelo's theorem that the cardinality of any model of ZFC2 is an inaccessible cardinal.

My question is whether there is a way of making a theory categorical that doesn't rely on Zermelo's theorem, and thus might work for theories that were not extensions of ZFC2.

More specifically, I want to know whether the union of $\omega$-many $V_{\kappa}$ for $\kappa$ inaccessible (i.e. the countably-infinite union of domains of models of ZFC2) could be categorically characterized. Clearly, the union is not itself a model of ZFC2 since it has countable cofinality. Hence, we cannot use the "extension of ZFC2" method mentioned above.

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Answering your more specific question: yes, it$^*$ can be so characterized, in the following way:

  • First, start with the axioms of second-order Zermelo set theory with choice (without Replacement). Note that $V_\mu$ is a model of this if $\mu$ is the first limit of inaccessibles - indeed, as long as $\lambda$ is a limit ordinal greater than $\omega$ we have that $V_\lambda$ satisfies second-order Zermelo set theory with choice.

  • Next, add "For every ordinal $\alpha$ there is an inaccessible cardinal $>\alpha$."

  • EDIT: Now, add "Every set is contained in a set model of ZFC$_2$." Note that since ZFC$_2$ has only finitely many axioms, this is in fact expressible by a single second-order sentence. This axiom gives "local replacement" - in particular, it implies that $V_\alpha$ exists for each ordinal $\alpha$ in the model.

  • Finally, add "There is no ordinal which is a limit of inaccessibles."

This characterizes $V_\mu$ up to isomorphism.


$^*$Or rather, the union of the first $\omega$-many $V_\kappa$s with $\kappa$ inaccessible is so characterizable; in general, if $(\kappa_i)_{i\in\omega}$ is an increasing sequence of inaccessibles there is of course no reason to believe that $\bigcup_{i\in\omega} V_{\kappa_i}=V_{\sup\{\kappa_i:i\in\omega\}}$ is so characterizable.

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  • $\begingroup$ You probably meant "the limit of the first $\omega$ inaccessibles" rather than "the $\omega$th inaccessible. Also, the question (read literally) is about an unspecified $\omega$-cofinal limit of inaccessibles. The first of those won't satisfy "there is an ordinal which is a limit of inaccessibles." And if (like me) you believe that there are lots of inaccessible, then not all of their $\omega$-cofinal limits will have categorical descriptions, because there aren't enough theories to characterize them. $\endgroup$ – Andreas Blass Oct 31 '17 at 21:52
  • $\begingroup$ @AndreasBlass Whoops. Fixed (and of course I meant "there is no ..."). $\endgroup$ – Noah Schweber Oct 31 '17 at 22:42
  • $\begingroup$ I'm trying to give a formal proof that the theory (call it T) given by @NoahSchweber does categorically determine the union. Begin by assuming two models N and M of T, and show (I presume) that the domains of N, M are in fact identical with the specified union. But I'm having trouble showing that the union is a subset of N and M. It's evident that N and M will have the same inaccessible "height" $\kappa$, but what's to prevent them from differing below $\kappa$? $\endgroup$ – Mallik Nov 2 '17 at 3:13
  • $\begingroup$ @Mallik Try proving that their respective cumulative hierarchies agree "level-by-level" by induction on the level - that is, show by induction on $\alpha<\kappa$ that $V_\alpha^M=V_\alpha^N$. $\endgroup$ – Noah Schweber Nov 2 '17 at 3:16
  • $\begingroup$ Thanks. That's what I was thinking...i.e. show that the first limit of inaccessibles is the first place that categoricity could fail. But if I can't use Zermelo's categoricity theorems (because we're not using full ZFC2) can I formulate the induction hypothesis that the levels are isomorphic, indeed identical? $\endgroup$ – Mallik Nov 2 '17 at 3:18

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