Timeline for What are some proofs of Godel's Theorem which are *essentially different* from the original proof?
Current License: CC BY-SA 3.0
10 events
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| Nov 24, 2021 at 13:13 | comment | added | AnduinWilde | I would like to mention a proof based on Chatin's theorem and idea of "unexpected examination": ams.org/notices/201011/rtx101101454p.pdf | |
| Aug 7, 2011 at 16:42 | comment | added | Ali Enayat | 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). | |
| Aug 6, 2011 at 22:02 | comment | added | Ron Maimon | 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. | |
| Aug 5, 2011 at 22:29 | comment | added | Alon Amit | An awesome summary! | |
| Aug 5, 2011 at 20:54 | history | edited | Ron Maimon | CC BY-SA 3.0 |
Yes, I mean prove-ably,phone!
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| Aug 5, 2011 at 20:41 | history | edited | Ron Maimon | CC BY-SA 3.0 |
Not at all! Ah community wiki.
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| Aug 5, 2011 at 18:54 | comment | added | Ron Maimon | 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. | |
| Aug 5, 2011 at 15:02 | comment | added | Andreas Blass | 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. | |
| Aug 5, 2011 at 11:19 | history | edited | Kaveh | CC BY-SA 3.0 |
added sections to make the answer more readable
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| Aug 5, 2011 at 6:09 | history | answered | Ron Maimon | CC BY-SA 3.0 |