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A result on implications of Σ1-sentences and its application to normal form theorems

Published online by Cambridge University Press:  12 March 2014

Jean-Yves Girard  and
Peter Päppinghaus
Jean-Yves Girard
Affiliation:
Universität Hannover, D-3000 Hannover, Federal Republic of Germany
Peter Päppinghaus
Affiliation:
Université Paris VII, 75221 Paris, Cedex 05, France
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Extract

In this paper we are concerned with the formal provability of certain true implications of Σ1-sentences. Old completeness and incompleteness results already give some information about this. For example by Σ1-completeness of PRA (primitive recursive arithmetic) every true implication of the form D → E, where D is a Σ0-sentence and E a Σ1-sentence, or D a Σ1-sentence and E a true Σ1-sentence, is provable in PRA. On the other hand, by Gödel's incompleteness theorems one can define for every suitable theory S a false Σ1-sentence Ds such that for every false Σ0-sentence E the true implication Ds → E is not provable in S, but is provable in PRA + Cons. So one sees that for a suitable fixed conclusion the provability of true implications of Σ1-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 Σ1-sentences depends on the conclusion. As remarked above, by Σ1-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 Σ1-sentence, S an extension of PRA, and S+ ≔ PRA + IAΣ1 + RFNΣ1S). Depending on D and S one can define a Σ1-sentence Es such that S+ ⊢ D → Es, but S ⊬ D → Es provided that S+ is not strong enough to refute D. (IAΣ1 denotes the scheme of the induction axiom for Σ1-formulas, and RFNΣ1(S) the uniform Σ1-reflection principle for S.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 3 , September 1981 , pp. 634 - 642
Copyright
Copyright © Association for Symbolic Logic 1981

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References

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