16 March 2023, The Utrecht Logic in Progress Series (TULIPS), Jaap van Oosten

Abstract: Starting with a bit of history of the notion of PCAs (or “Schönfinkel Algebras”), we examine their role as building blocks of various notions of realizability. An extremely important construction is that of the “realizability topos RT(A)” on a PCA A. Immediately, notions from pure topos theory start to find their way into the study of PCAs. It will be argued that PCAs stand on the threshold between topos theory and abstract higher order computability theory (à la Longley-Normann). Particularly forceful is the work of Takeyuki Kihara.