Syntactical truth predicates for second order arithmetic

Abstract

AbstractWe introduce a notion of syntactical truth predicate (s.t.p.) for the second order arithmetic PA2. An s.t.p. is a set T of closed formulas such that:(i) T(t = u) if and only if the closed first order terms t and u are convertible, i.e., have the same value in the standard interpretation(ii) T(A → B) if and only if (T(A) ⇒ T(B))(iii) T(∀xA) if and only if (T(A[x ← t]) for any closed first order term t)(iv) T(∀X A) if and only if (T(A[X ← ∆]) for any closed set definition ∆ = {x ∣ D(x)}).S.t.p.'s can be seen as a counterpart to Tarski's notion of (model-theoretical) validity and have main model properties. In particular, their existence is equivalent to the existence of an ω-model of PA2, this fact being provable in PA2 with arithmetical comprehension only.