Logic Seminar

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.