Editing Gödel's incompleteness theorems (section)

==== Wittgenstein ====
[[Ludwig Wittgenstein]] wrote several passages about the incompleteness theorems that were published posthumously in his 1953 ''[[Remarks on the Foundations of Mathematics]]'', particularly, one section sometimes called the "notorious paragraph" where he seems to confuse the notions of "true" and "provable" in Russell's system. Gödel was a member of the [[Vienna Circle]] during the period in which Wittgenstein's early [[ideal language philosophy]] and [[Tractatus Logico-Philosophicus]] dominated the circle's thinking. There has been some controversy about whether Wittgenstein misunderstood the incompleteness theorem or just expressed himself unclearly. Writings in Gödel's [[Nachlass]] express the belief that Wittgenstein misread his ideas.

Multiple commentators have read Wittgenstein as misunderstanding [[Gödel]], although {{harvtxt|Floyd|Putnam|2000}} as well as {{harvtxt|Priest|2004}} have provided textual readings arguing that most commentary misunderstands Wittgenstein.{{sfnm
 | 1a1 = Rodych | 1y = 2003
 | 2a1 = Floyd | 2a2 = Putnam | 2y = 2000
 | 3a1 = Priest | 3y = 2004
}} On their release, Bernays, Dummett, and Kreisel wrote separate reviews on Wittgenstein's remarks, all of which were extremely negative.{{sfn|Berto|2009|p=208}} The unanimity of this criticism caused Wittgenstein's remarks on the incompleteness theorems to have little impact on the logic community. In 1972, Gödel stated: "Has Wittgenstein lost his mind? Does he mean it seriously? He intentionally utters trivially nonsensical statements", and wrote to [[Karl Menger]]  that Wittgenstein's comments demonstrate a misunderstanding of the incompleteness theorems writing:

<blockquote> It is clear from the passages you cite that Wittgenstein did ''not'' understand [the first incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics).{{sfn|Wang|1996|p=179}} </blockquote>

Since the publication of Wittgenstein's ''Nachlass'' in 2000, a series of papers in philosophy have sought to evaluate whether the original criticism of Wittgenstein's remarks was justified. {{harvtxt|Floyd|Putnam|2000}} argue that Wittgenstein had a more complete understanding of the incompleteness theorem than was previously assumed. They are particularly concerned with the interpretation of a Gödel sentence for an ω-inconsistent system as saying "I am not provable", since the system has no models in which the provability predicate corresponds to actual provability. {{harvtxt|Rodych|2003}} argues that their interpretation of Wittgenstein is not historically justified. {{harvtxt|Berto|2009}} explores the relationship between Wittgenstein's writing and theories of paraconsistent logic.{{sfnm
 | 1a1 = Floyd | 1a2 = Putnam | 1y = 2000
 | 2a1 = Rodych | 2y = 2003
 | 3a1 = Berto | 3y = 2009
}}