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Aug 10, 2016 at 14:52 vote accept C. Eratosthene
Jun 13, 2016 at 20:06 history edited Drike CC BY-SA 3.0
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Jun 13, 2016 at 20:04 comment added Drike @godelian: thanks man! Edited accordingly.
Jun 13, 2016 at 20:03 comment added godelian Drike: Gödel's original proof does make use of König's lemma, which is a weak form of choice. For the case of a countable language, we now know that such an assumption is not needed.
Jun 13, 2016 at 20:00 history edited Drike CC BY-SA 3.0
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Jun 13, 2016 at 19:53 history edited Drike CC BY-SA 3.0
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Jun 13, 2016 at 19:46 history edited Drike CC BY-SA 3.0
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Jun 13, 2016 at 19:35 comment added Drike I thought you did indeed, but was not quite sure anymore!
Jun 13, 2016 at 16:41 comment added Joel David Hamkins I think I gave those arguments here: mathoverflow.net/a/46729/1946.
Jun 13, 2016 at 16:09 history edited Drike CC BY-SA 3.0
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Jun 13, 2016 at 15:58 comment added Drike The fundamental Theorem of Algebra (stating that $\bf C$ is algebraically closed) also has proofs of very different nature. One proof that a field has a unique algebraic closure uses Gödel's Compactness Theorem, both for existence and unicity. I believe I saw these proofs here on MO.
Jun 13, 2016 at 15:54 history edited Drike CC BY-SA 3.0
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Jun 13, 2016 at 15:46 history edited Drike CC BY-SA 3.0
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Jun 13, 2016 at 15:42 comment added Joel David Hamkins For compactness, there is also the Henkin proof and the proof from Boolean-valued models.
Jun 13, 2016 at 15:38 comment added Drike The Banach-Tarski paradox has also several proofs.
S Jun 13, 2016 at 15:36 history answered Drike CC BY-SA 3.0
S Jun 13, 2016 at 15:36 history made wiki Post Made Community Wiki by Drike