zbMATH — the first resource for mathematics

On the Herbrand notion of consistency for finitely axiomatizable fragments of bounded arithmetic theories. (English) Zbl 1099.03050
The main result proved in the paper says that there exists a number \(n\) such that \(\bigcup_{m} S_{m}\) (the union of the bounded arithmetic theories \(S_{m}\)) does not prove the Herbrand consistency of the finitely axiomatizable theory \(S^{n}_{3}\). This is proved by using a modification of methods applied by Z. Adamowicz in the paper “Herbrand consistency and bounded arithmetic” [Fundam. Math. 171, 279–292 (2002; Zbl 0995.03044)].

03F30 First-order arithmetic and fragments
Full Text: DOI
[1] DOI: 10.1007/s001530000072 · Zbl 1030.03043
[2] DOI: 10.4064/fm171-3-7 · Zbl 0995.03044
[3] On tableaux consistency in weak theories (2001)
[4] How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson’s arithmetic Q 67 pp 465– (2002) · Zbl 1004.03050
[5] Logic, Methodology, and Philosophy of Science VIII (Moscow 1987) pp 143– (1989)
[6] DOI: 10.1016/0168-0072(94)00049-9 · Zbl 0834.03022
[7] Fundamenta Mathematicae 136 pp 85– (1990)
[8] Logic colloquium ’03 24 pp 244– (2006)
[9] Cuts, consistency statements, and interpretations 50 pp 423– (1985)
[10] Bounded Arithmetic, Propositional Logic, and Complexity Theory (1995) · Zbl 0835.03025
[11] Metamathematics of First-Order Arithmetic (1993) · Zbl 0781.03047
[12] DOI: 10.1016/0168-0072(87)90066-2 · Zbl 0647.03046
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.