zbMATH — the first resource for mathematics

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.
03F30 First-order arithmetic and fragments
03F20 Complexity of proofs
Full Text: DOI
[1] Annals of the Japan Association for Philosophy of Science 6 pp 195– (1984) · Zbl 0545.03032
[2] Tsukuba Journal of Mathematics 2 pp 69– (1978)
[3] DOI: 10.1090/S0002-9947-1973-0432416-X
[4] Arithmetic, proof theory, and computational complexity (Papers from the Conference Held in Prague, July 2–5, 1991) 23 pp 30– (1993)
[5] DOI: 10.1007/BF01625836 · Zbl 0644.03032
[6] Theories very close to PA where Kreisel’s conjecture is false 72 pp 123– (2007) · Zbl 1117.03065
[7] One hundred and two problems in mathematical logic 40 pp 113– (1975)
[8] Tsukuba Journal of Mathematics 4 pp 115– (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.