Turing Uncomputability

Jørgen Veisdal in Privatdozent:

As Soare (2013) recounts, John von Neumann (1903–1957) happened to be in the audience as a representative of Hilbert’s program when Gödel, then 25 years old, took the podium to present his result. von Neumann immediately recognized that Hilbert’s program was over, and spent the next weeks preparing the proof of a related theorem. He had in mind an arithmetization of Gödel’s incompleteness result, to show that not only are formal systems incapable of proving every statement in them, they are also unable to guarantee proofs of their own consistency. He later presented his proof to Gödel, writing “using the methods you employed so successfully […] I achieved a result that seems to me to be remarkable, namely, I was able to to show that the consistency of mathematics is unprovable” (Dyson, 2005). Writing back, Gödel reportedly politely thanked the great man and informed him that he himself (Gödel) had written the same proof weeks earlier, and that it had already been submitted for publication.

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