Re:Kurt Gödel’s Brilliant Madness
by ProfHewitt on Thursday April 15, 2021 @07:41PM
(#26827537)
Actually, [Gödel 1931] claimed that the incompleteness result
was for Russell's Principia Mathematica, which is much
stronger than a first-order theory of the natural numbers.
Also, Gödel's incompleteness result was not for the liar paradox.
Instead, it was for the proposition I'mUnprovable, which
[Gödel 1931] claimed to have constructed. However,
[Wittgenstein 1937-1944] showed that I'mUnprovable leads to
inconsistency in the foundations of mathematics. Fortunately,
I'mUnprovable cannot be constructed in foundations because
the attempted construction violates order on propositions.
Furthermore, Gödel did not have a proof of the second
incompleteness result at the time he wrote to von Neumann [von Plato 2018].
von Neumann never realized the deception by Gödel because it
was discovered after both had died.
> Kurt Gödel, only a year beyond his PhD, announced a result which would forever change the foundations of mathematics. He formalized the liar paradox, "This statement is false" to prove roughly that for any effectively axiomatized consistent extension T of number theory (Peano arithmetic) there is a sentence σ which asserts its own unprovability in T. John von Neumann, who was in the audience immediately understood the importance of Gödel's incompleteness theorem. [...]
> In the next few weeks von Neumann realized that by arithmetizing the proof of Gödel's first theorem, one could prove an even better one, that no such formal system T could prove its own consistency. A few weeks later he brought his proof to Gödel, who thanked him and informed him politely that he had already submitted the second incompleteness theorem for publication.