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Introduction Foundations of Mathematics

  • Yvon Gauthier
Chapter
Part of the Synthese Library book series (SYLI, volume 310)

Abstract

Foundational questions in mathematics were born with Hilbert, but foundational programs existed before him. Arithmetization of analysis and arithmetization of algebra (for Kronecker) antidate Hilbert’s idea of axiomatization. While Frege was struggling with the logical concept of number as the extension of a concept and while Cantor (and Dedekind) imagined infinite (transfinite) extensions of the ordinary number concept. Kronecker was busy devising a general arithmetic that would arithmetize mathematics without transcending the realm of the algebraic. The so-called foundational crisis did affect only the logicist program, comforting in a sense the arithmetical program. It is that program that Hilbert wanted to pursue with other means in order to rescue set theory from its logico-paradoxical consequences.

Keywords

Proof Theory General Arithmetic Peano Arithmetic Recursion Theory Compact Cardinal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Yvon Gauthier
    • 1
  1. 1.University of MontréalCanada

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