David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
In Zettel, Wittgenstein considered a modified version of Cantor’s diagonal argument. According to Wittgenstein, Cantor’s number, different with other numbers, is defined based on a countable set. If Cantor’s number belongs to the countable set, the definition of Cantor’s number become incomplete. Therefore, Cantor’s number is not a number at all in this context. We can see some examples in the form of recursive functions. The definition "f(a)=f(a)" can not decide anything about the value of f(a). The definiton is incomplete. The definition of "f(a)=1+f(a)" can not decide anything about the value of f(a) too. The definiton is incomplete.
According to Wittgenstein, the contradiction, in Cantor's proof, originates from the hidden presumption that the definition of Cantor’s number is complete. The contradiction shows that the definition of Cantor’s number is incomplete.
According to Wittgenstein’s analysis, Cantor’s diagonal argument is invalid. But different with Intuitionistic analysis, Wittgenstein did not reject other parts of classical mathematics. Wittgenstein did not reject definitions using self-reference, but showed that this kind of definitions is incomplete.
Based on Thomson’s diagonal lemma, there is a close relation between a majority of paradoxes and Cantor’s diagonal argument. Therefore, Wittgenstein’s analysis on Cantor’s diagonal argument can be applied to provide a unified solution to paradoxes.
|Keywords||Cantor’s diagonal argument Wittgenstein self-reference paradox|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Matthew E. Moore (2002). A Cantorian Argument Against Infinitesimals. Synthese 133 (3):305 - 330.
R. T. Brady & P. A. Rush (2008). What is Wrong with Cantor's Diagonal Argument? Logique Et Analyse 51 (1):185-219..
Marcus Rossberg & Philip A. Ebert (2010). Cantor on Frege's Foundations of Arithmetic : Cantor's 1885 Review of Frege's Die Grundlagen der Arithmetik. History and Philosophy of Logic 30 (4):341-348.
Christopher Menzel (1984). Cantor and the Burali-Forti Paradox. The Monist 67 (1):92-107.
William Boos (1987). Consistency and Konsistenz. Erkenntnis 26 (1):1 - 43.
Vojtěch Kolman (2010). Continuum, Name and Paradox. Synthese 175 (3):351 - 367.
Anne Newstead (2009). Cantor on Infinity in Nature, Number, and the Divine Mind. American Catholic Philosophical Quarterly 83 (4):533-553.
Added to index2010-05-01
Total downloads78 ( #16,453 of 1,098,129 )
Recent downloads (6 months)3 ( #112,729 of 1,098,129 )
How can I increase my downloads?