論理的意味論と整合性証明

概要

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We consider the philosophical and historical significance of the consistency problem of mathematical systems, from the view point of logical semantics theories. In §1 we present a traditional problem which is related to the consistency problem, and discuss Leibniz's attitude to this traditional problem, from the view point of his contextual theory of meaning. In §2 we see that the tranditional problem is identified with the consistency problem in Hilbert's verification theory of meaning. In §3 we see that Husserl's treatment on the manifold-axiom system relation can be interpreted in the frame-work of the sematics-syntax relation in the sense of Tarski's meaning theory, and how Husserl's solution for the traditional problem is related to the consistency problem. In §4 we present Gentzen's standpoint which is based on a verification theory of meaning, and discuss the philosophical significance of his consistency proof of number theory from the view point of his meaning theory. In §5 we present Wittgenstein's criticism on the consistency problem from the conventionalist view, and reanalize the philosophical significance of Gentzen's consistency proof through the Wittgenstein's criticism.