Journal of Symbolic Logic

Herbrand consistency of some arithmetical theories

Saeed Salehi

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Abstract

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 IΔ0+Ωm 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⊇ IΔ0+Ω2 in T itself.

In this paper, the above results are generalized for IΔ0+Ω1. 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 IΔ0+Ω1 and IΔ0.

Article information

Source
J. Symbolic Logic, Volume 77, Issue 3 (2012), 807-827.

Dates
First available in Project Euclid: 13 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1344862163

Digital Object Identifier
doi:10.2178/jsl/1344862163

Mathematical Reviews number (MathSciNet)
MR2987139

Zentralblatt MATH identifier
1256.03065

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

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

Citation

Salehi, Saeed. Herbrand consistency of some arithmetical theories. J. Symbolic Logic 77 (2012), no. 3, 807--827. doi:10.2178/jsl/1344862163. https://projecteuclid.org/euclid.jsl/1344862163


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