Realizability Models Separating Various Fan Theorems

Abstract

Abstract. We develop a realizability model in which the realizers are the reals not just Turing computable in a fixed real but rather the reals in a countable ideal of Turing degrees. This is then applied to prove several separation results involving variants of the Fan Theorem.

Separating Fragments of Wlem, Lpo, and Mp

“…What has been proved [33] is that WWKL is strictly weaker than WKL over RCA 0 , a weak (classical) subsystem of second order arithmetic. That proof was then adapted to a constructive setting in [16], but applied to the classical contrapositives of WKL and WWKL, there called D-FAN and W-D-FAN, respectively. It is then shown that the Yu-Simpson argument that WWKL does not imply WKL translates to a proof that over IZF W-D-FAN does not imply D-FAN.…”