Wittgenstein And Labyrinth Of ‘Actual Infinity’: The Critique Of Transfinite Set Theory

Abstract

In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory (including the Diagonal Argument, the Continuum Hypothesis and Cantor’s Theorem) and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing (completed) infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems.