Wittgenstein’s ‘notorious paragraph’ about the Gödel Theorem

In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” in Gödel’s proof, as Floyd and Putnam suggest (Floyd (2000), Floyd (2001)), or whether this remark reveals Wittgenstein’s misunderstanding of Gödel’s proof as Rodych and Steiner argued for recently (Rodych (1999, 2002, 2003), Steiner (2001)). In the following the arguments of both interpretations will be sketched and some deficiencies will be identified. Afterwards a detailed reconstruction of Wittgenstein’s argument will be offered. It will be seen that Wittgenstein’s argumentation is meant to be a rejection of Gödel’s proof but that it cannot satisfy this pretension

A general principle for purely model-theoretical proofs of Gödel’s second incompleteness theorem

By generalizing Kreisel’s proof of the Second Incompleteness Theorem of Gödel I extract a general principle which can also be used for other purely model-theoretical proofs of that theorem

Wittgenstein´s Critique of Gödel´s Incompleteness Results

It is often said that Gödel´s famous theorem of 1931 is\ud equal to the Cretian Liar, who says that everything that he\ud says is a lie. But Gödel´s result is only similar to this\ud sophism and not equivalent to it. When mathematicians\ud deal with Gödel´s theorem, then it is often the case that\ud they become poetical or even emotional: some of them\ud show a high esteem of it and others despise it. Wittgenstein\ud sees the famous Liar as a useless language game\ud which doesn´t excite anybody. Gödel´s first incompleteness\ud theorem shows us that in mathematics there are\ud puzzles which have no solution at all and therefore in\ud mathematics one should be very careful when one\ud chooses a puzzle on which one wants to work. Gödel´s\ud second imcompleteness theorem deals with hidden\ud contradictions – Wittgenstein shows a paradigmatic\ud solution: he simply shrugs his shoulders on this problem\ud and many mathematicians do so today as well. Wittgenstein\ud says than Gödel´s results should not be treated as\ud mathematical theorems, but as elements of the humanistic\ud sciences. Wittgenstein sees them as something which\ud should be worked on in a creative manner

Consequences of a Goedel's misjudgment

The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce dangerous fruits, as to apply the incompleteness Theorems to the full second-order Arithmetic and to deduce the semantic incompleteness of its language by these same Theorems. The first three paragraphs are introductory and serve to define the languages inherently semantic and its properties, to discuss the consequences of the expression order used in a language and some question about the semantic completeness: in particular is highlighted the fact that a non-formal theory may be semantically complete despite using a language semantically incomplete. Finally, an alternative interpretation of the Goedel's unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog