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Syntactical truth predicates for second order arithmetic

  • Loïc Colson (a1) and Serge Grigorieff (a2)
Abstract
Abstract

We 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(AB) if and only if (T(A) ⇒ T(B))

(iii) T(∀xA) if and only if (T(A[xt]) for any closed first order term t)

(iv) T(∀X A) if and only if (T(A[X ← ∆]) for any closed set definition ∆ = {xD(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.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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