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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)].

##### MSC:
 03F30 First-order arithmetic and fragments
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##### References:
 [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
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