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Mathematical significance of consistency proofs

Published online by Cambridge University Press:  12 March 2014

G. Kreisel
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Extract

The principal aim of the present paper is to sketch some mathematical applications of the work of the Hilbert school in the foundations of mathematics. Most of them depend on the ε-theorems of Hilbert-Bernays II [6], or of Ackermann [1], but instead the work of Gentzen [5] or Schütte [25], which does not use the ε-symbol, could have been applied.

The avowed purpose of all this work is described in the introduction to volume I of Hilbert-Bernays: the consistency of the usual principles of proof is to be established by means of finitist methods, or at least, by means of methods which are more “evident” or more “constructive” than the principles under discussion. This formulation seems to have several defects:

  • (1) Since the notion of constructive proof is vague, the whole formulation of the program is vague; and though an exact formulation constitutes, of course, an interesting problem for the logician, because of this vagueness the mathematician does not find the program attractive.

  • (2) The formulation does not cover too well the actual substance of the material contained in [6]; e.g. the ε-theorems for the predicate calculus go far beyond establishing mere consistency of the predicate calculus. It is significant that it is precisely these theorems which lead to the more interesting applications, e.g. the use of the ε-theorems with equality in the solution of Hilbert's 17th problem given below.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 23 , Issue 2 , June 1958 , pp. 155 - 182
Copyright
Copyright © Association for Symbolic Logic 1958

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References

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