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Computability Learning Seminar

Tuesday, November 12, 2019 — 2:00 PM EST

Dino Rossegger, Department of Pure Mathematics, University of Waterloo

"Determinacy in second order arithmetic"

We stick with the theme of the previous seminar sessions and continue our introduction of reverse mathematics. We pause with our reading of Slicing the truth'' and consider a different flavor of reverse mathematics: determinacy of games in $2^\omega$ and $\omega^\omega$. Let $\Gamma$ be a collection of sets of real numbers. $\Gamma$-determinacy is the statement that for every $A\in \Gamma$, one of the players in the 2 player perfect information game for $A$ has a winning strategy. Since the beginnings of the study of modern set theory the study of determinacy statements has been an important part of mathematical logic. In fact, Friedman's result that $\Sigma^0_5-determinacy is not provable in second order arithmetic is widely considered to be the starting point of reverse mathematics. In this talk we will give an overview about determinacy statements and results about determinacy in reverse mathematics. If time allows we will prove two theorems by Montalb\'an and Shore: that$\mathbb{ACA}_0\vdash \Delta^0_3$-TD$ and that $\mathbb{RCA}_0\not\vdash \Delta^0_3-TD$ where $\Gamma-TD$ is the statement that Turing invariant sets in $\Gamma$ are determined.

MC 5413

May 2021

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