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Syntactical truth predicates for second order arithmetic

Published online by Cambridge University Press:  12 March 2014

Loïc Colson  and
Serge Grigorieff
Loïc Colson
Affiliation:
L.L.A.I.C., UFR d'Informatique, Université Paris7, 2 Place Jussieu 75251 Paris, France, E-mail: colson@lita.univ-metz.fr
Serge Grigorieff
Affiliation:
L.L.A.I.C. UFR d'Informatique, Université Paris 7, 2 Place Jussieu 75251 Paris, France, E-mail: seg@ufr-info-p7.jussieu.fr
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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(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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 1 , March 2001 , pp. 225 - 256
Copyright
Copyright © Association for Symbolic Logic 2001

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