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    $\begingroup$ Your criterion, that two proofs are the same if they give the same construction, is very restrictive. Consider, for example, the well-known proof that there are infinitely many primes, the proof where you multiply the first $n$ primes, add 1, and find a prime factor of the result. Now modify it by changing "add 1" to "subtract 1". The modification results in finding a different prime. Yet most mathematicians would not consider it a really different proof. You probably intended something like "the same construction up to silly changes", but it's not easy to define silliness. $\endgroup$
    – Andreas Blass
    Commented Aug 5, 2011 at 15:02
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    $\begingroup$ Of course you are right. The way I think of silly changes is by the complexity of the proof required to prove statement II given statement I and vice versa. For the example you gave, I would be happy thinking of them as (slightly)different proofs because to get from one to the other is not much simpler than proving either. There is a measure of closeness defined by how long/complex (axiom strength wise) the equivalence between the constructions is. $\endgroup$
    – Ron Maimon
    Commented Aug 5, 2011 at 18:54
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    $\begingroup$ An awesome summary! $\endgroup$
    – Alon Amit
    Commented Aug 5, 2011 at 22:29
  • $\begingroup$ I still am having some trouble with the full computational interpretation of Jech/Woodin. The simpler consequences are easy enough to interpret as standard type I arguments, but there is one theorem which is completely different: there is no descending infinite sequence of models of set theory. I had a similar proof for the well-foundedness of the collection of theories stronger than PA under the ordering A is stronger than B when A proves the consistency of B. But this theorem has a more involved proof than type I arguments. I'll try to finish Jech Woodin today. $\endgroup$
    – Ron Maimon
    Commented Aug 6, 2011 at 22:02
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    $\begingroup$ The Jech/Woodin proof has an important ancestor, due to Kreisel, who came up with the first model-theoretic proof of the second incompleteness theorem in the 1960's (see, e.g., logika.umk.pl/llp/06/du.pdf). $\endgroup$
    – Ali Enayat
    Commented Aug 7, 2011 at 16:42