Set-theoretic potentialism and the universal finite set, Scandinavian Logic Symposium, June 2018

This will be an invited talk at the Scandinavian Logic Symposium SLS 2018, held at the University of Gothenburg in Sweden, June 11-13, 2018.

Abstract. Providing a set-theoretic analogue of the universal algorithm, 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 sufficiently large $V_\theta$; the set is empty in any transitive model; 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 $N$ of ZFC in which $\varphi$ defines the new set $z$. I shall draw out consequences of the universal finite set for set-theoretic potentialism and discuss several issues it raises in the philosophy of set theory.

The talk will include joint work with W. Hugh Woodin, Øystein Linnebo and others.

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