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Great read. > Kurt Gödel, only a year beyond his PhD, announced a result which would forever change the foundations of mathematics. He formalized the liar para
by lqet 6y ago
Great read.
> 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.
- ProfHewitt 6y agoActually, [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.