Editing Gödel's incompleteness theorems (section)

== History ==
After Gödel published his proof of the [[completeness theorem]] as his doctoral thesis in 1929, he turned to a second problem for his [[habilitation]]. His original goal was to obtain a positive solution to [[Hilbert's second problem]].{{sfn|Dawson|1997|p=63}} At the time, theories of natural numbers and real numbers similar to [[second-order arithmetic]] were known as "analysis", while theories of natural numbers alone were known as "arithmetic".

Gödel was not the only person working on the consistency problem. [[Wilhelm Ackermann|Ackermann]] had published a flawed consistency proof for analysis in 1925, in which he attempted to use the method of [[Epsilon calculus|ε-substitution]] originally developed by Hilbert. Later that year, [[John von Neumann|von Neumann]] was able to correct the proof for a system of arithmetic without any axioms of induction. By 1928, Ackermann had communicated a modified proof to Bernays; this modified proof led Hilbert to announce his belief in 1929 that the consistency of arithmetic had been demonstrated and that a consistent proof of analysis would likely soon follow. After the publication of the incompleteness theorems showed that Ackermann's modified proof must be erroneous, von Neumann produced a concrete example showing that its main technique was unsound.{{sfnm
 | 1a1 = Zach | 1y = 2007 | 1p = 418
 | 2a1 = Zach | 2y = 2003 | 2p = 33
}}

In the course of his research, Gödel discovered that although a sentence, asserting its falsehood leads to paradox, a sentence that asserts its non-provability does not. In particular, Gödel was aware of the result now called [[Tarski's indefinability theorem]], although he never published it. Gödel announced his first incompleteness theorem to Carnap, Feigel, and Waismann on August 26, 1930; all four would attend the [[Second Conference on the Epistemology of the Exact Sciences]], a key conference in [[Königsberg]] the following week.

=== Announcement ===
The 1930 [[Second Conference on the Epistemology of the Exact Sciences|Königsberg conference]] was a joint meeting of three academic societies, with many of the key logicians of the time in attendance. Carnap, Heyting, and von Neumann delivered one-hour addresses on the mathematical philosophies of logicism, intuitionism, and formalism, respectively.{{sfn|Dawson|1996|p=69}} The conference also included Hilbert's retirement address, as he was leaving his position at the University of Göttingen. Hilbert used the speech to argue his belief that all mathematical problems can be solved. He ended his address by saying,
{{Blockquote|For the mathematician there is no ''[[Ignoramus et ignorabimus|Ignorabimus]]'', and, in my opinion, not at all for natural science either. ... The true reason why [no one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolish ''Ignorabimus'', our credo avers: We must know. We shall know!}}
This speech quickly became known as a summary of Hilbert's beliefs on mathematics (its final six words, "''Wir müssen wissen. Wir werden wissen!''", were used as Hilbert's epitaph in 1943). Although Gödel was likely in attendance for Hilbert's address, the two never met face to face.{{sfn|Dawson|1996|p=72}}

Gödel announced his first incompleteness theorem at a roundtable discussion session on the third day of the conference. The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for a conversation. Later that year, working independently with knowledge of the first incompleteness theorem, von Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated November 20, 1930.{{sfn|Dawson|1996|p=70}} Gödel had independently obtained the second incompleteness theorem and included it in his submitted manuscript, which was received by ''Monatshefte für Mathematik'' on November 17, 1930.

Gödel's paper was published in the ''Monatshefte'' in 1931 under the title "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("[[On Formally Undecidable Propositions in Principia Mathematica and Related Systems I]]"). As the title implies, Gödel originally planned to publish a second part of the paper in the next volume of the ''Monatshefte''; the prompt acceptance of the first paper was one reason he changed his plans.{{sfn|van Heijenoort|1967|loc=page 328, footnote 68a}}

=== Generalization and acceptance ===

Gödel gave a series of lectures on his theorems at Princeton in 1933–1934 to an audience that included Church, Kleene, and Rosser. By this time, Gödel had grasped that the key property his theorems required is that the system must be effective (at the time, the term "general recursive" was used). Rosser proved in 1936 that the hypothesis of ω-consistency, which was an integral part of Gödel's original proof, could be replaced by simple consistency if the Gödel sentence was changed appropriately. These developments left the incompleteness theorems in essentially their modern form.

Gentzen published his [[Gentzen's consistency proof|consistency proof]] for first-order arithmetic in 1936. Hilbert accepted this proof as "finitary" although (as Gödel's theorem had already shown) it cannot be formalized within the system of arithmetic that is being proved consistent.

The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof of the incompleteness theorems in the second volume of ''Grundlagen der Mathematik'' ([[#{{harvid|Bernays|1939}}|1939]]), along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first full published proof of the second incompleteness theorem.

=== 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
}}