This will be a talk for the European Set Theory Conference 2022 in Turin, Italy 29 August – 2 September 2022.

**Abstract.** Every countable model of ZFC set theory with an inner model satisfying a sufficient theory must also have an end-extension satisfying that theory. For example, every countable model with a measurable cardinal has an end-extension to a model of $V=L[\mu]$; every model with extender-based large cardinals has an end-extension to a model of $V=L[\vec E]$; every model with infinitely many Woodin cardinals and a measurable above has an end-extension to a model of $\text{ZF}+\text{DC}+V=L(\mathbb{R})+\text{AD}$. These results generalize the famous Barwise extension theorem, of course, asserting that every countable model of ZF set theory admits an end-extension to a model of $\text{ZFC}+{V=L}$, a theorem which was simultaneously a technical culmination of Barwise’s pioneering methods in admissible set theory and infinitary logic and also one of those rare mathematical theorems that is saturated with philosophical significance. In this talk, I shall describe a new proof of the Barwise theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory, while also providing the generalization I mentioned. This proof furthermore leads directly to the universal finite sequence, a $\Sigma_1$-definable finite sequence, which can be extended arbitrarily as desired in suitable end-extensions of the universe, a result holding important consequences for the nature of set-theoretic potentialism. This work is joint with Kameryn J. Williams.

- J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” Journal of Symbolic Logic, 2021.

[Bibtex]`@ARTICLE{HamkinsWilliams2021:The-universal-finite-sequence, author = {Joel David Hamkins and Kameryn J. Williams}, title = {The $\Sigma_1$-definable universal finite sequence}, journal = {Journal of Symbolic Logic}, year = {2021}, volume = {}, number = {}, pages = {}, month = {}, note = {}, abstract = {}, keywords = {}, eprint = {1909.09100}, archivePrefix = {arXiv}, primaryClass = {math.LO}, source = {}, doi = {10.1017/jsl.2020.59}, }`