In several publications, Juliet Floyd and Hilary Putnam have argued that the so-called ‘notorious paragraph’ of the Remarks on the Foundations of Mathematics contains a valuable philosophical insight about Gödel’s informal proof of the first incompleteness theorem – in a nutshell, the idea they attribute to Wittgenstein is that if the Gödel sentence of a system is refutable, then, because of the resulting ω-inconsistency of the system, we should give up the translation of Gödel’s sentence by the English sentence “I am unprovable”.
I will argue against Floyd and Putnam’s use of the idea, and I will indirectly question its attribution to Wittgenstein. First, I will point out that the idea is inefficient in the context of the first incompleteness theorem because there is an explicit assumption of soundness in Gödel’s informal discussion of that theorem. Secondly, I will argue that of he who makes the observation that Floyd and Putnam think Wittgenstein has made about the first theorem, one will expect to see an analogous observation (concerning the ‘consistency’ statement of systems) about Gödel’s second incompleteness theorem – yet we see nothing to that effect in Wittgenstein’s remarks. Incidentally, that never-made remark on the import of the second theorem is of genuine logical significance.
Floyd, Juliet & Putnam, Hilary: A note on Wittgenstein’s ‘notorious paragraph’ about the Gödel theorem, in: The Journal of Philosophy, 97 (2000), 624 – 632.10.5840/jphil2000971129Search in Google Scholar
Floyd, Juliet & Putnam, Hilary: Bays, Steiner, and Wittgenstein's ‘notorious’ paragraph about the Gödel theorem, in: The Journal of Philosophy, 103 (2006), 101 – 110.10.5840/jphil2006103228Search in Google Scholar
Gödel, Kurt: On formally undecidable propositions of Principia Mathematica and related systems I (1931), in: Solomon Feferman et al. (eds.): Kurt Gödel Collected Works, Volume I: Publications 1929 – 1936, Oxford 1986, 145 – 195.Search in Google Scholar
Isaacson, Daniel: Necessary and sufficient conditions for undecidability of the Gödel sentence and its truth, in: David DeVidi et al. (eds.): Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L. Bell, Dordrecht 2011, 135 – 152.10.1007/978-94-007-0214-1_7Search in Google Scholar
Putnam, Hilary: A note on Steiner on Wittgenstein, Gödel, and Tarski, in: Iyyun: The Jerusalem Philosophical Quarterly, 57 (2008), 83 – 93.Search in Google Scholar
Rodych, Victor: Misunderstanding Gödel: New arguments about Wittgenstein and new remarks by Wittgenstein, in: Dialectica, 57 (2003), 279 – 313.10.1111/j.1746-8361.2003.tb00272.xSearch in Google Scholar
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