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  1. Abelian groups and quadratic residues in weak arithmetic.Emil Jeřábek - 2010 - Mathematical Logic Quarterly 56 (3):262-278.
    We investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Specifically, we show that the structure theorem for finite abelian groups is provable in S22 + iWPHP, and use it to derive Fermat's little theorem and Euler's criterion for the Legendre symbol in S22 + iWPHP extended by the pigeonhole principle PHP. We prove the quadratic reciprocity theorem in the arithmetic theories T20 + Count2 and I Δ0 + Count2 with modulo-2 counting (...)
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  • Finitistic Arithmetic and Classical Logic.Mihai Ganea - 2014 - Philosophia Mathematica 22 (2):167-197.
    It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition. The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's ‘predicative arithmetic’ program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly (...)
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  • Quadratic forms in models of IΔ0+ Ω1, Part II: Local equivalence.Paola D’Aquino & Angus Macintyre - 2011 - Annals of Pure and Applied Logic 162 (6):447-456.
    In this second paper of the series we do a local analysis of quadratic forms over completions of a non-standard model of IΔ0+Ω1.
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