Abstract
We are familiar with various set-theoretical paradoxes such as Cantor's paradox, Burali-Forti's paradox, Russell's paradox, Russell-Myhill paradox and Kaplan's paradox. In fact, there is another new possible set-theoretical paradox hiding itself in Wittgenstein’s Tractatus (Wittgenstein 1989). From the Tractatus’s Picture theory of language (hereafter LP) we can strictly infer the two contradictory propositions simultaneously: (a) the world and the language are equinumerous; (b) the world and the language are not equinumerous. I call this antinomy the world-language paradox. Based on a rigorous analysis of the Tractatus, with the help of the technical resources of Cantor’s naive set theory (Cantor in Mathematische Annalen, 46, 481–512, 1895, Mathematische Annalen, 49, 207–246, 1897) and Zermelo-Fraenkel set theory with the axiom of choice (hereafter ZFC) (Jech 2006: 3–15; Kunen 1992: xv–xvi; Bagaria 2008: 619–622), I outline the world-language paradox and assess the unique possible solution plan, i.e., the mathematical plan utilizing ‘infinity’. I conclude that Wittgenstein cannot solve the hidden set-theoretical paradox of the Tractatus successfully unless he gives up his finitism.
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Notes
‘The New Wittgenstein’ is a new way of interpreting Wittgenstein, inter alia, the Tractatus. It can be traced back to the 1960s, but becomes popular mainly after 2000. Its main representatives are Cora Diamond, James Conant and Warren Goldfarb in the USA. They generally advocate an ‘austere’ and ‘resolute’ reading of the Tractatus, especially §6.54, and believe that: (1) the Tractatus does not ‘show’ any ‘unsayable’ metaphysics about the world, the language or the logic; (2) all of the Tractatus’s propositions are ‘nonsensical’; and (3) the Tractatus is deeply consistent with the later Wittgenstein in respect of some non-constructive ‘therapeutic’ philosophical purport (cf. Crary & Read. 2000: 1–18; Hacker 2003: 1–4; Goldfarb 2011: 7; Diamond 2015: 1–2).
We should recall ‘Hilbert’s Infinite Hotel’ (cf. Byers 2007: 161–163).
We must remember: one of Cantor’s greatest contributions to the mathematics or the philosophy consists in his convincing demonstration that ‘infinity has different sizes’.
It must be noted that in fact, Wittgenstein believes that we cannot state the number of the formal concepts such as the objects, the facts and the propositions. Talking about the number of all objects, for instance saying ‘there are 100 objects’ or saying ‘there are ℵ 0 objects’, is utterly nonsensical (cf. §4.1272).
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Li, J. The Hidden Set-Theoretical Paradox of the Tractatus . Philosophia 46, 159–164 (2018). https://doi.org/10.1007/s11406-017-9904-2
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DOI: https://doi.org/10.1007/s11406-017-9904-2