Endextensions of models of set theory and the Sigma_1 universal finite sequence
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.
[Slides]
Recall that if $M \subseteq N$ are models of set theory then $N$ endextends $M$ 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 endextension $N \models \overline{\mathsf{ZF}}$ of $M$ in which the sequence is exactly $t$.

Indeed, for the previous statements it suffices that $M \models \mathsf{ZF}$ and endextends 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 endextensional settheoretic 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 endextensional 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 rankextensions of Hamkins and Woodin, and both are cousins of Woodin’s universal algorithm for arithmetic.