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

Ithaque 10:43-65 (2012)
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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 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 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

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Valérie Lynn Therrien
McGill University

Citations of this work

Wittgenstein on Set Theory and the Enormously Big.Ryan Dawson - 2015 - Philosophical Investigations 39 (4):313-334.

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References found in this work

Philosophical Investigations.Ludwig Wittgenstein - 1953 - New York, NY, USA: Wiley-Blackwell. Edited by G. E. M. Anscombe.
Remarks on the foundations of mathematics.Ludwig Wittgenstein - 1967 - Oxford [Eng.]: Blackwell. Edited by G. E. M. Anscombe, Rush Rhees & G. H. von Wright.
Philosophical grammar.Ludwig Wittgenstein - 1974 - Oxford [Eng.]: Blackwell. Edited by Rush Rhees.
Remarks on the Foundations of Mathematics.Ludwig Wittgenstein - 1956 - Oxford: Macmillan. Edited by G. E. M. Anscombe, Rush Rhees & G. H. von Wright.
Philosophical remarks.Ludwig Wittgenstein - 1975 - Chicago: University of Chicago Press. Edited by Rush Rhees.

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