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Logic Seminar

Wednesday, October 20, 2021
12:00pm to 1:00pm
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The axiom of real determinacy in admissible sets
Juan P. Aguilera, Department of Mathematics, Ghent University,

The axiom of real determinacy (ADR) asserts the determinacy of every infinite two-player, perfect-information game with moves in the set of real numbers. By a theorem of Woodin, ZF+ADR is consistent if and only if ZFC is consistent together with the existence of a cardinal λ which is a limit of Woodin cardinals and <λ-strong cardinals.
In this talk, we explore the strength of ADR over the theory KP+"R exists" and observe that it is much weaker. Indeed, the theory KP+ADR+"R exists" is weaker than ZFC+"there are ω2 Woodin cardinals". This is a consequence of the following theorem: over ZFC, the existence of a transitive model of KP+ADR containing the set of all real numbers is equivalent to the determinacy of all open games of length ω3.

For more information, please email A. Kechris at kechris@caltech.edu.