This will be a talk for the CUNY Logic Workshop, November 17, 2017, 2pm GC Room 6417.

**Abstract.** I shall define a certain finite set in set theory $$\{x\mid\varphi(x)\}$$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition $\varphi$ has complexity $\Sigma_2$ and therefore any instance of it $\varphi(x)$ is locally verifiable inside any sufficient $V_\theta$; the set is empty in any transitive model and others; and if $\varphi$ defines the set $y$ in some countable model $M$ of ZFC and $y\subset z$ for some finite set $z$ in $M$, then there is a top-extension of $M$ to a model $N$ in which $\varphi$ defines the new set $z$. In particular, although there are models of set theory with maximal $\Sigma_2$ theories, nevertheless no model of set theory realizes a maximal $\Sigma_2$ theory with its natural-number parameters. Using the universal finite set, it follows that the validities of top-extensional set-theoretic potentialism, the modal principles valid in the Kripke model of all countable models of set theory, each accessing its top-extensions, are precisely the assertions of S4. Furthermore, if ZFC is consistent, then there are models of ZFC realizing the top-extensional maximality principle.

This is joint work with W. Hugh Woodin.

This is a very interesting result. Though not in the same flavor, but maybe the following result of Gitik-shelah be also interesting. By a result of Hajnal, and independently Shelah, the set $\{ \lambda^\delta: 2^\delta < \lambda \}$ is always finite. By the work of Gitik-Shelah, given any natural number $n \geq 2$, this set can have exactly $n$ members in a suitable generic extension (assuming the existence of enough strong cardinals).

Mohammad, could you please add some references to the mentioned results for the sake of completeness?

Thanks in advance

See the paper “On certain indestructibility of strong cardinals and a question of Hajnal”:

https://link.springer.com/article/10.1007%2FBF01624081

Thank you very much!