The Sigma_1 universal finite sequence
This is a contributed talk at the seventh biannual European Set Theory Confence on Thursday, 4 July 2019. It is about joint work with Joel David Hamkins and Philip Welch.
[slides]
Consider a computably enumerable theory $\mathsf{ZF}^+$ extending $\mathsf{ZF}$. There is a $\Sigma_1$ definition, without parameters, for a finite sequence of sets $b_0, b_1, \ldots, b_n$ with the following properties.

$\mathsf{ZF}^+$ proves the sequence is finite.

In any transitive model of $\mathsf{ZF}^+$ the sequence is empty.

If $M$ is a countable model of $\mathsf{ZF}^+$ in which the sequence is $s$ and $t \in M$ is a finite sequence extending $s$, then there is $N \models \mathsf{ZF}^+$ endextending $M$—that is, $N$ extends $M$ and if $x \in M$, $y \in N$, and $N \models y \in x$ then $y \in M$—so that in $N$ the sequence is $t$.

Indeed, it suffices merely that $M \models \mathsf{ZF}$ and there is an inner model $W \subseteq M$ with $W \models \mathsf{ZF}^+$.
This $\Sigma_1$ universal finite sequence can be compared to the $\Sigma_2$ universal finite set of Hamkins and Woodin. Their finite set is universal with respect to rankextensions, with $\Sigma_2$ statements being those witnessed by a rankinitial segment of the universe. The universal sequence presented here is universal with respect to endextensions and is thus $\Sigma_1$, as $\Sigma_1$ statements are those witnessed by an $\in$initial segment of the universe, i.e. by a transitive set. Both universal sequences are set theoretic analogues of Woodin’s universal algorithm for models of arithmetic.
As a consequence of the $\Sigma_1$ universal finite sequence we can calculate the modal validities of endextensional set theoretic potentialism. Recall that a potentialist system is a Kripke model of firstorder structures in a common extension, where the accessibility relation refines the substructure relation. A potentialist system thereby has a natural associated modal interpretation, namely where $\possible\varphi$ holds in a world $M$ if $\varphi$ holds in some extension of $M$ and $\necessary\varphi$ holds in $M$ if $\varphi$ holds in every extension of $M$. Consider the potentialist system consisting of countable models of $\mathsf{ZF}^+$ ordered by endextension. Its modal validities are precisely $\mathsf{S4}$.