We prove that the Gödel incompleteness theorem holds for a weak arithmetic Tₘ = IΔ₀ + Ωₘ, for m ≥ 2, in the form Tₘ ⊬ HCons(Tₘ), where HCons(Tₘ) is an arithmetic formula expressing the consistency of Tₘ with respect to the Herbrand notion of provability. Moreover, we prove , where is HCons relativised to the definable cut Iₘ of (m-2)-times iterated logarithms. The proof is model-theoretic. We also prove a certain non-conservation result for Tₘ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-3-7,
author = {Zofia Adamowicz},
title = {Herbrand consistency and bounded arithmetic},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {279-292},
zbl = {0995.03044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-3-7}
}
Zofia Adamowicz. Herbrand consistency and bounded arithmetic. Fundamenta Mathematicae, Tome 173 (2002) pp. 279-292. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-3-7/