This will be a talk for the CUNY Logic Workshop, November 17, 2017, 2pm GC Room 6417. 

Abstract. I shall define a certain finite set in set theory {𝑥∣𝜑⁡(𝑥)} and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition 𝜑 has complexity Σ2 and therefore any instance of it 𝜑⁡(𝑥) is locally verifiable inside any sufficient 𝑉𝜃; the set is empty in any transitive model and others; and if 𝜑 defines the set 𝑦 in some countable model 𝑀 of ZFC and 𝑦 ⊂𝑧 for some finite set 𝑧 in 𝑀, then there is a top-extension of 𝑀 to a model 𝑁 in which 𝜑 defines the new set 𝑧. In particular, although there are models of set theory with maximal Σ2 theories, nevertheless no model of set theory realizes a maximal Σ2 theory with its natural-number parameters. Using the universal finite set, it follows that the validities of top-extensional set-theoretic potentialism, the modal principles valid in the Kripke model of all countable models of set theory, each accessing its top-extensions, are precisely the assertions of S4. Furthermore, if ZFC is consistent, then there are models of ZFC realizing the top-extensional maximality principle.

This is joint work with W. Hugh Woodin.