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Abstract: I shall define a certain finite set in set theory $\{x\mid\varphi(x)\}$ 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 \(\varphi\) has complexity \(\Sigma_2\) and therefore any instance of it \(\varphi(x)\) is locally verifiable inside any sufficient \(V_\theta\); the set is empty in any transitive model and others; and if \(\varphi\) defines the set \(y\) in some countable model \(M\) of ZFC and \(y\subset z\) for some finite set \(z\) in \(M\), then there is a top-extension of \(M\) to a model \(N\) in which \(\varphi\) 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.