Editing Gödel's incompleteness theorems (section)

=== Criticisms ===

==== Finsler ====
{{harvtxt|Finsler|1926}} used a version of [[Richard's paradox]] to construct an expression that was false but unprovable in a particular, informal framework he had developed.{{sfn|Finsler|1926}} Gödel was unaware of this paper when he proved the incompleteness theorems (Collected Works Vol. IV., p.&nbsp;9). Finsler wrote to Gödel in 1931 to inform him about this paper, which Finsler felt had priority for an incompleteness theorem. Finsler's methods did not rely on formalized provability and had only a superficial resemblance to Gödel's work.{{sfn|van Heijenoort|1967|p=328}} Gödel read the paper but found it deeply flawed, and his response to Finsler laid out concerns about the lack of formalization.{{sfn|Dawson|1996|p=89}} Finsler continued to argue for his philosophy of mathematics, which eschewed formalization, for the remainder of his career.

==== Zermelo ====
<!-- Zermelo criticized the incompleteness theorems for their reliance on finitary proofs. In 1931, he presented a system of infinitely-long proofs, which he believed could be used to overcome the limitations shown by the incompleteness theorems.-->
In September 1931, [[Ernst Zermelo]] wrote to Gödel to announce what he described as an "essential gap" in Gödel's argument.{{sfn|Dawson|1996|p=76}} In October, Gödel replied with a 10-page letter, where he pointed out that Zermelo mistakenly assumed that the notion of truth in a system is definable in that system; it is not true in general by [[Tarski's undefinability theorem]].{{sfnm
 | 1a1 = Dawson | 1y = 1996 | 1p = 76
 | 2a1 = Grattan-Guinness | 2y = 2005 | 2pp = 512–513
}} However, Zermelo did not relent and published his criticisms in print with "a rather scathing paragraph on his young competitor".{{sfn|Grattan-Guinness|2005|pp=513}} Gödel decided that pursuing the matter further was pointless, and Carnap agreed.{{sfn|Dawson|1996|p=77}} Much of Zermelo's subsequent work was related to logic stronger than first-order logic, with which he hoped to show both the consistency and categoricity of mathematical theories.

==== 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
}}