## Journal of Symbolic Logic

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

### Herbrand consistency of some arithmetical theories

#### 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 IΔ

_{0}+Ω

_{m}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⊇ IΔ

_{0}+Ω

_{2}in T itself.

In this paper, the above results are generalized for
IΔ_{0}+Ω_{1}. Also after tailoring the definition of Herbrand
consistency for IΔ_{0} we prove the corresponding theorems for
IΔ_{0}. Thus the Herbrand version of Gödel's second
incompleteness theorem follows for the theories IΔ_{0}+Ω_{1}
and IΔ_{0}.

#### 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