k-Provability in $$\hbox {PA}$$

Author:

Santos Paulo Guilherme,Kahle Reinhard

Abstract

AbstractWe study the decidability of k-provability in $$\hbox {PA}$$ PA —the relation ‘being provable in $$\hbox {PA}$$ PA with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton (the list of axioms and rules that were used). The decidability of k-provability for the usual Hilbert-style formalisation of $$\hbox {PA}$$ PA is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms.

Funder

Eberhard Karls Universität Tübingen

Publisher

Springer Science and Business Media LLC

Subject

Applied Mathematics,Logic

Reference12 articles.

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1. k-Provability in PA;2021-10-06

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