This will be a talk for the Rutgers Logic Seminar, April 2, 2018. Hill Center, Busch campus.

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.  The definition can be thought of as an idealized diamond sequence, and there are consequences for the philosophical theory of set-theoretic top-extensional potentialism.

This is joint work with W. Hugh Woodin.

• The universal finite set
• The modal logic of arithmetic potentialism and the universal algorithm