A Philosophical Significance of Gentzen’s 1935 Consistency Proof for First-Order Arithmetic:

on a Circularity of Implication

Abstract

    Gentzen remarked that one of the aims of his 1935/36 consistency proofs for first-order arithmetic was to give a “finitist” interpretation for the implication-formulas in first-order arithmetic. He imposed the following requirement on such an interpretation: a “finitist” interpretation for the implication-formulas must be able to avoid the circularity of implication that was urged by himself. However, Gentzen did not present his “finitist” interpretation explicitly. Moreover, he gave no argument for its non-circularity. In this paper, first we formulate an interpretation for the implication-formulas in first-order arithmetic by using Gentzen’s 1935 consistency proof. Next, we argue that this interpretation avoids the circularity urged by Gentzen.