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The Internal Consistency of Arithmetic with Infinite Descent: A Syntactical Proof

  • Yvon Gauthier
Chapter
  • 631 Downloads
Part of the Studies in Universal Logic book series (SUL)

Abstract

The question of the consistency or non-contradiction of arithmetic is a philosophical question, that is the certainty of a mathematical theory and it has become a logical problem requiring a mathematical solution. It is Hilbert who has put the original question and who has attempted a first answer. Having demonstrated in 1899 the consistency of elementary geometry on the basis of a consistent arithmetic of real numbers, he turned to the question of that arithmetic (the second problem of his 1900 list) which included besides the axiom of elementary arithmetic an axiom of continuity, i.e. the Archimedean axiom with syntactic completeness. Hilbert introduces functionals, that is second-order functions or functions of functions, in order the use induction over the ordinals. Transfinite induction proved successful in the hands of Gentzen, and thereafter Gödel’s second incompleteness results which aimed to show that Peano’s arithmetic cannot contain its own consistency.

Keywords

Homogeneous Polynomial Finite Type Double Negation General Arithmetic Consistency Proof 
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 International Publishing Switzerland 2015

Authors and Affiliations

  • Yvon Gauthier
    • 1
  1. 1.University of MontrealMontrealCanada

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