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 $$\{x\mid \phi (x)\}$$ 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 $\phi$ has complexity ${\mathrm{\Sigma }}_2$ and therefore any instance of it $\phi (x)$ is locally verifiable inside any sufficient $V_{\theta }$; the set is empty in any transitive model and others; and if $\phi$ 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 $\phi$ defines the new set $z$. In particular, although there are models of set theory with maximal ${\mathrm{\Sigma }}_2$ theories, nevertheless no model of set theory realizes a maximal ${\mathrm{\Sigma }}_2$ 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.