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

Ithaque 10:43-65 (2012)

Authors
Valérie Lynn Therrien
McGill University
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
Keywords Wittgenstein  Set theory  Philosophy of Mathematics  Diagonal Proof  Infinity  Continuum Hypothesis  Cantor's Theorem
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References found in this work BETA

Philosophical Remarks.Guy Stock - 1976 - Philosophical Quarterly 26 (103):178-180.
Philosophical Grammar.Ludwig Wittgenstein, Rush Rhees & Anthony Kenny - 1975 - Philosophy and Rhetoric 8 (4):260-262.
Remarks on the Foundations of Mathematics.Alice Ambrose - 1957 - Philosophy and Phenomenological Research 18 (2):262-265.

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Citations of this work BETA

Wittgenstein on Set Theory and the Enormously Big.Ryan Dawson - 2016 - Philosophical Investigations 39 (4):313-334.

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