Mathematical significance of consistency proofs

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Abstract
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 theε-theorems of Hilbert-Bernays II [6], or of Ackermann [1], but instead the work of Gentzen [5] or Schütte [25], which does not use theε-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. theε-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 theε-theoremswithequality in the solution of Hilbert's 17th problem given below.

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