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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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