Editing Gödel's incompleteness theorems (section)

==== 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.