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benelliott comments on Edward Nelson claims proof of inconsistency in Peano Arithmetic - Less Wrong
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From what gathered both went down a similar route at the same time, although it is possible that I am mistaken in this. I named Cauchy as he seems to be the best-known of the two, and therefore the one with the highest absolutely-positively-not-just-some-crank factor.
If I lose the bet (not that anyone has yet agreed to accept the bet), then payment will be based on restricted number theory without the axiom of induction.