Analiza i Egzystencja


ISSN: 1734-9923     eISSN: 2300-7621    OAI    DOI: 10.18276/aie.2023.63-03
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Issue archive / 63 (2023)
Gramatyka nieskończoności. Ludwiga Wittgensteina krytyka teorii mnogości
(Grammar of Infinity. Ludwig Wittgenstein's Critique of Set Theory)

Authors: Piotr Dehnel ORCID
Pomorska Szkoła Wyższa w Starogardzie Gdańskim
Keywords: Cantor Dedekind language set theory infinity diagonal proof.
Data publikacji całości:2023
Page range:33 (55-87)
Cited-by (Crossref) ?:

Abstract

The paper discusses a relatively underexamined element of Wittgenstein’s philosophy of mathematics associated with his critique of set theory. I outline Wittgenstein’s objections to the theories of Dedekind and Cantor, including the confounding of extension and intension, the faulty definition of the infinite set as infinite extension and the critique of Cantor’s diagonal proof. One of Wittgenstein’s major objections to set theory was that the concept of the size of infinite sets, which Cantor expressed by means of symbols אₒ and c, had no application, i.e. that there was no grammatical technique that could show how such expressions were to be used. Notions of set theory are, so to speak, exterior – they find themselves outdoors, outside of what we usually do. They form a discourse that takes us beyond the horizon of everydayness and commonality. They are like an engine idling of the language of mathematics.
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