# Seminars & Colloquia Calendar

## The Universal Finite SetÂ

#### 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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Paul Feehan, Daniel Ketover, Natasa Sesum Organizer's webpage

Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward Organizer's webpage

Jason Saied Seminar webpage

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Quentin Dubroff Organizer's webpage

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Edna Jones Organizer's webpage

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Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum Organizer's Webpage

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Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi Organizer's webpage

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Joel Lebowitz, Michael Kiessling

Yanyan Li, Haim Brezis Organizer's Webpage

Stephen D. Miller, John C. Miller, Alex V. Kontorovich, Alex Walker seminar website

Stephen D. Miller

Brooke Ogrodnik, Website

Organizers: Yanyan Li, Z.C. Han, Jian Song, Natasa Sesum

Yael Davidov Seminar webpage

Kristen Hendricks, Xiaochun Rong, Hongbin Sun, Chenxi Wu Organizer's page

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Ebru Toprak, Organizer

Organizer's webpage: Organizer's webpage

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