CERES: An analysis of Fürstenberg's proof of the infinity of primes

  • Authors:

  • Matthias Baaz;Stefan Hetzl;Alexander Leitsch;Clemens Richter;Hendrik Spohr

  • Affiliations:

  • Institute of Discrete Mathematics and Geometry (E104), Vienna University of Technology, Wiedner Hauptstraβe 8-10, 1040 Vienna, Austria;Institute of Computer Languages (E185), Vienna University of Technology, Favoritenstraβe 9, 1040 Vienna, Austria;Institute of Computer Languages (E185), Vienna University of Technology, Favoritenstraβe 9, 1040 Vienna, Austria;Institute of Computer Languages (E185), Vienna University of Technology, Favoritenstraβe 9, 1040 Vienna, Austria;Institute of Computer Languages (E185), Vienna University of Technology, Favoritenstraβe 9, 1040 Vienna, Austria

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2008

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Abstract

The distinction between analytic and synthetic proofs is a very old and important one: An analytic proof uses only notions occurring in the proved statement while a synthetic proof uses additional ones. This distinction has been made precise by Gentzen's famous cut-elimination theorem stating that synthetic proofs can be transformed into analytic ones. CERES (cut-elimination by resolution) is a cut-elimination method that has the advantage of considering the original proof in its full generality which allows the extraction of different analytic arguments from it. In this paper we will use an implementation of CERES to analyze Furstenberg's topological proof of the infinity of primes. We will show that Euclid's original proof can be obtained as one of the analytic arguments from Furstenberg's proof. This constitutes a proof-of-concept example for a semi-automated analysis of realistic mathematical proofs providing new information about them.