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Wednesday, October 20, 2021
12:00 PM - 1:00 PM
Online Event

# Logic Seminar

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.