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},
  }