September 2012 Herbrand consistency of some arithmetical theories
Saeed Salehi
J. Symbolic Logic 77(3): 807-827 (September 2012). DOI: 10.2178/jsl/1344862163


Gödel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171 (2002), pp. 279—292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories IΔ0m with m ≥ 2, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory T⊇ IΔ02 in T itself.

In this paper, the above results are generalized for IΔ01. Also after tailoring the definition of Herbrand consistency for IΔ0 we prove the corresponding theorems for IΔ0. Thus the Herbrand version of Gödel's second incompleteness theorem follows for the theories IΔ01 and IΔ0.


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Saeed Salehi. "Herbrand consistency of some arithmetical theories." J. Symbolic Logic 77 (3) 807 - 827, September 2012.


Published: September 2012
First available in Project Euclid: 13 August 2012

zbMATH: 1256.03065
MathSciNet: MR2987139
Digital Object Identifier: 10.2178/jsl/1344862163

Primary: Primary 03F40, 03F30; Secondary 03F05, 03H15

Keywords: bounded arithmetics, weak arithmetics , Cut-free provability , Gödel's Second Incompleteness Theorem , Herbrand provability

Rights: Copyright © 2012 Association for Symbolic Logic


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Vol.77 • No. 3 • September 2012
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