# Seminars & Colloquia Calendar

Logic Seminar

## The Universal Finite Set

#### Joel Hamkins -- CUNY

Location:  Hill 705
Date & time: Monday, 02 April 2018 at 5:00PM - 6:00PM

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.

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