Determinacy in third order arithmetic

In recent work, Schweber [7] introduces a framework for reverse mathematics in a third order setting and investigates several natural principles of transfinite recursion. The main result of that paper is a proof, using the method of forcing, that in the context of two-person perfect information games with moves in R, open determinacy (Σ1 -DET) is not implied by clopen determinacy (∆ R 1 -DET). In this paper, we give another proof of this result by isolating a level of L witnessing this separation. We give a notion of β-absoluteness in the context of third-order arithmetic, and show that this level of L is a β-model; combining this with our previous results in [2], we show that Σ04-DET, determinacy for games on ω with Σ04 payoff, is sandwiched between Σ R 1 -DET and ∆ R 1 -DET in terms of β-consistency strength. §