What he could have said (but did not say) about Gödel’s second theorem: A note on Floyd-Putnam’s Wittgenstein

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. ‏ ‎This short paper is a by-product of the lecture I gave, as an invited speaker, at the Fourth Annual Conference of the Iranian Association for Logic, 2016. I am grateful to Saeed Salehi for an ongoing and productive discussion on different aspects of Gödel’s 1931 paper, and to Ali Masoudi and Mousa Mohammadian for all the friendly and brotherly support. I’d like to dedicate this paper to the memory of my teacher, John V. Canfield (1934 – 2017).