Showing papers by "Jean-Yves Girard published in 1981"

A result on implications of σ1-sentences and its application to normal form theorems

Abstract: In this paper we are concerned with the formal provability of certain true implications of Σ 1 -sentences. Old completeness and incompleteness results already give some information about this. For example by Σ 1 -completeness of PRA (primitive recursive arithmetic) every true implication of the form D → E , where D is a Σ 0 -sentence and E a Σ 1 -sentence, or D a Σ 1 -sentence and E a true Σ 1 -sentence, is provable in PRA. On the other hand, by Godel's incompleteness theorems one can define for every suitable theory S a false Σ 1 -sentence D s such that for every false Σ 0 -sentence E the true implication D s → E is not provable in S , but is provable in PRA + Con s . So one sees that for a suitable fixed conclusion the provability of true implications of Σ 1 -sentences depends on the content of the premise. Now we ask, if and how for a suitable fixed premise the provability of true implications of Σ 1 -sentences depends on the conclusion. As remarked above, by Σ 1 -completeness of PRA this question is settled, if the premise is true. For a false premise it is answered in § 1 as follows: Let D be a false Σ 1 -sentence, S an extension of PRA, and S + ≔ PRA + IA Σ 1 + RFN Σ 1 S). Depending on D and S one can define a Σ 1 -sentence E s such that S + ⊢ D → E s , but S ⊬ D → E s provided that S + is not strong enough to refute D . (IA Σ 1 denotes the scheme of the induction axiom for Σ 1 -formulas, and RFN Σ 1 (S) the uniform Σ 1 -reflection principle for S .)