The Hidden Set-Theoretical Paradox of the Tractatus

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.