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Kreisel’s conjecture with minimality principle. (English) Zbl 1180.03054
Summary: In the paper a theory PA\(_{M}\) is considered – it is an axiomatization of Peano arithmetic in which the minimality principle is used instead of the scheme of induction and identity is finitely axiomatized using identity axioms of the form \(x=y \rightarrow S(x)=S(y)\) for function symbols of PA. The author proves Kreisel’s Conjecture for this system. It is shown that the result is independent of the choice of language of PA\(_{M}\). The author proves also that if infinitely many instances of \(A(x)\) are provable in a bounded number of steps in PA\(_{M}\) then there exists \(k \in \omega\) such that PA\(_{M} \vdash \forall x > k A(x)\). The results of the paper imply that PA\(_{M}\) does not prove the scheme of induction or identity schemes in a bounded number of steps.
MSC:
03F30 First-order arithmetic and fragments
03F20 Complexity of proofs
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References:
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