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  1. On the Concept of Finitism.Luca Incurvati - 2015 - Synthese 192 (8):2413-2436.
    At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.
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  • William Tait. The Provenance of Pure Reason. Essays on the Philosophy of Mathematics and on its History.Charles Parsons - 2009 - Philosophia Mathematica 17 (2):220-247.
    William Tait's standing in the philosophy of mathematics hardly needs to be argued for; for this reason the appearance of this collection is especially welcome. As noted in his Preface, the essays in this book ‘span the years 1981–2002’. The years given are evidently those of publication. One essay was not previously published in its present form, but it is a reworking of papers published during that period. The Introduction, one appendix, and some notes are new. Many of the essays (...)
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  • Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on Finitism, Constructivity and Hilbert's Program.Solomon Feferman - 2008 - Dialectica 62 (2):179-203.
    This is a survey of Gödel's perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program, using his published and unpublished articles and lectures as well as the correspondence between Bernays and Gödel on these matters. There is also an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end.
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  • Unfolding Finitist Arithmetic.Solomon Feferman & Thomas Strahm - 2010 - Review of Symbolic Logic 3 (4):665-689.
    The concept of the (full) unfolding of a schematic system is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted ? The program to determine for various systems of foundational significance was previously carried out for a system of nonfinitist arithmetic, ; it was shown that is proof-theoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system (...)
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  • Herbrand’s Fundamental Theorem in the Eyes of Jean Van Heijenoort.Claus-Peter Wirth - 2012 - Logica Universalis 6 (3-4):485-520.
    Using Heijenoort’s unpublished generalized rules of quantification, we discuss the proof of Herbrand’s Fundamental Theorem in the form of Heijenoort’s correction of Herbrand’s “False Lemma” and present a didactic example. Although we are mainly concerned with the inner structure of Herbrand’s Fundamental Theorem and the questions of its quality and its depth, we also discuss the outer questions of its historical context and why Bernays called it “the central theorem of predicate logic” and considered the form of its expression to (...)
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  • Consistency, Models, and Soundness.Matthias Schirn - 2010 - Axiomathes 20 (2-3):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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