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 ; every model with extender-based large cardinals has an end-extension to a model of ; every model with infinitely many Woodin cardinals and a measurable above has an end-extension to a model of . 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 , 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 -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.
