This is a talk in the virtual CUNY Models of Peano Arithmetic seminar, on 2020 July 29. It is about joint work with Joel David Hamkins. Our paper can be found here.

Recall that if $M \subseteq N$ are models of set theory then $N$ end-extends $M$ if $N$ does not have new elements for sets in $M$. In this talk I will discuss a $\Sigma_1$-definable finite sequence which is universal for end extensions in the following sense. Consider a computably axiomatizable extension $\overline{\mathsf{ZF}}$ of $\mathsf{ZF}$. There is a $\Sigma_1$-definable finite sequence

$a_0, a_1, \ldots, a_n$

with the following properties.

  • $\mathsf{ZF}$ proves that the sequence is finite.

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

  • If $M$ is a countable model of $\overline{\mathsf{ZF}}$ in which the sequence is $s$ and $t \in M$ is a finite sequence extending $s$ then there is an end-extension $N \models \overline{\mathsf{ZF}}$ of $M$ in which the sequence is exactly $t$.

  • Indeed, for the previous statements it suffices that $M \models \mathsf{ZF}$ and end-extends a submodel $W \models \overline{\mathsf{ZF}}$ of height at least $(\omega_1^{\mathrm{L}})^M$.

This universal finite sequence can be used to determine the modal validities of end-extensional set-theoretic potentialism, namely to be exactly the modal theory $\mathsf{S4}$. The sequence can also be used to show that every countable model of set theory extends to a model satisfying the end-extensional maximality principle, asserting that any possibly necessary sentence is already true.

The $\Sigma_1$ universal finite sequence is a sister to the $\Sigma_2$ universal finite sequence for rank-extensions of Hamkins and Woodin, and both are cousins of Woodin’s universal algorithm for arithmetic.

Extending the universal finite sequence by moving to an end-extension.