Joint with Joel David Hamkins.
Abstract We introduce the $\Sigma_1$-definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\Sigma_1$-definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if $M$ is a countable model of set theory in which the sequence is $s$ and $t$ is any finite extension of $s$ in this model, then there is an end extension of $M$ to a model in which the sequence is $t$. Our proof method grows out of a new infinitary-logic- free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V = L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.
This project is a foray in the world of set-theoretic potentialism. The basic idea is, we conceive of the set theoretic multiverse as having a potentialist flavor; rather than there being a single fixed universe, there are multiple universes and from the current universe we can move to a bigger (better?) universe. The question then is, what is the flavor of this potentialist system? Are we building toward a well-defined single truth of set theory? Or is there branching in the truth possibilities, where we make permantent choices and seal off one possibility in favor of another?
We show that the potentialist system consisting of countable models of set theory ordered by end-extension, where $N \supseteq M$ is an end-extension if $M$ is closed $\in^N$-downward in $N$, has a branching flavor. Its modal validities are precisely the theory S4. Our main tool to show this is the titular $\Sigma_1$-definable universal finite sequence. As we move upward in this potentialist system, we commit to more and more of this universal sequence, thereby making permanent choices about $\Sigma_1$ truths.
The $\Sigma_1$-definable universal finite sequence is a sister to the Hamkins–Woodin $\Sigma_2$-definable universal finite set. Their universal finite set is universal with respect to rank-extensions, where all new sets have higher rank than any old set, whereas our universal finite sequence is universal with respect to end-extensions. Both are cousins to Woodin’s universal algorithm in arithmetic. And all three give rise to corresponding potentialist systems with S4 as their modal validities.