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  1. On the Intuitionistic Background of Gentzen's 1935 and 1936 Consistency Proofs and Their Philosophical Aspects.Yuta Takahashi - 2018 - Annals of the Japan Association for Philosophy of Science 27:1-26.
    Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer (...)
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  • Semantic Values for Natural Deduction Derivations.Göran Sundholm - 2006 - Synthese 148 (3):623-638.
    Drawing upon Martin-Löf’s semantic framework for his constructive type theory, semantic values are assigned also to natural-deduction derivations, while observing the crucial distinction between consequence among propositions and inference among judgements. Derivations in Gentzen’s format with derivable formulae dependent upon open assumptions, stand, it is suggested, for proof-objects, whereas derivations in Gentzen’s sequential format are proof-acts.
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  • Consistency, Models, and Soundness.Matthias Schirn - 2010 - Axiomathes 20 (2):153-207.
    This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. (...)
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  • Induction and transfinite induction in intuitionistic systems.Bruno Scarpellini - 1972 - Annals of Mathematical Logic 4 (2):173.
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  • Completeness and Hauptsatz for second order logic.Dag Prawitz - 1967 - Theoria 33 (3):246-258.
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  • A constructive consistency proof of a fragment of set theory.Jon Pearce - 1984 - Annals of Pure and Applied Logic 27 (1):25-62.
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  • Consistency of Heyting arithmetic in natural deduction.Annika Kanckos - 2010 - Mathematical Logic Quarterly 56 (6):611-624.
    A proof of the consistency of Heyting arithmetic formulated in natural deduction is given. The proof is a reduction procedure for derivations of falsity and a vector assignment, such that each reduction reduces the vector. By an interpretation of the expressions of the vectors as ordinals each derivation of falsity is assigned an ordinal less than ε 0, thus proving termination of the procedure.
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  • Proof-theoretical analysis: weak systems of functions and classes.L. Gordeev - 1988 - Annals of Pure and Applied Logic 38 (1):1-121.
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