The Nonarithmeticity of the Class of Realizable Predicate Formulas

Vaught's Theorem on Axiomatizability by a Scheme

“…, v k−1 ) and form n ( C ). 11 We find in V that, under interpretation L, we have SAT(n, B ). Let s be any empty set.…”

“…Variants of the same trick were used, after Vaught's paper, in other contexts, e.g., by Plisko, in his work on the predicate logic of realizability. See[11] and[12].…”

Generalized Realizability and Intuitionistic Logic

“…Constructive logics based on the recursive realizability essential differ from classical and intuitionistic logics. As an example, the predicate logic of recursive realizability is not recursively enumerable [19], while classical and intuitionistic predicate logics are conversely recursively enumerable. Various forms of subrecursive realizability were considered: primitive recursive realizability [1,24], minimal realizability [2], and corresponding predicate logics [15,17,25].…”

No Escape from Vardanyan's theorem

“…Vardanyan's work used earlier ideas of V.E. Plisko (see [Pli77], [Pli78], [Pli83], [Pli93]) and S. Artemov (see [Art85]). Vardanyan's Theorem was discovered independently but slightly later by Vann McGee.…”