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).

Does Choice Really Imply Excluded Middle? Part I: Regimentation of the Goodman–Myhill Result, and Its Immediate Reception†

ABSTRACT The one-page 1978 informal proof of Goodman and Myhill is regimented in a weak constructive set theory in free logic. The decidability of identities in general ($a\!=\!b\vee\neg a\!=\!b$) is derived; then, of sentences in general ($\psi\vee\neg\psi$). Martin-Löf’s and Bell’s receptions of the latter result are discussed. Regimentation reveals the form of Choice used in deriving Excluded Middle. It also reveals an abstraction principle that the proof employs. It will be argued that the Goodman–Myhill result does not provide the constructive set theorist with a dispositive reason for not adopting (full) Choice.

Visual thinking and simplicity of proof

This paper studies how spatial thinking interacts with simplicity in [informal] proof, by analysing a set of example proofs mainly concerned with Ferrers diagrams (visual representations of partitions of integers) and comparing them to proofs that do not use spatial thinking. The analysis shows that using diagrams and spatial thinking can contribute to simplicity by (for example) avoiding technical calculations, division into cases, and induction, and creating a more surveyable and explanatory proof (both of which are connected to simplicity). In response to one part of Hilbert's 24th problem, the area between two proofs is explored in one example, showing that between a proof that uses spatial reasoning and one that does not, there is a proof that is less simple yet more impure than either. This has implications for the supposed simplicity of impure proofs. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

Niven’s Theorem

Summary This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].

INFORMAL PROOF, FORMAL PROOF, FORMALISM

Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened.

Flexary Operations

In this article we introduce necessary notation and definitions to prove the Euler’s Partition Theorem according to H.S. Wilf’s lecture notes [31]. Our aim is to create an environment which allows to formalize the theorem in a way that is as similar as possible to the original informal proof. Euler’s Partition Theorem is listed as item #45 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [30].

Lagrange’s Four-Square Theorem

Summary This article provides a formalized proof of the so-called “the four-square theorem”, namely any natural number can be expressed by a sum of four squares, which was proved by Lagrange in 1770. An informal proof of the theorem can be found in the number theory literature, e.g. in [14], [1] or [23]. This theorem is item #19 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.