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There are so many applications of the Axiom of Choice (and consequently its equivalents) in number theory. But do you know any application of the Zorn's Lemma in Number Theory !? I mean a theorem or a problem in this field that uses Zorn's Lemma ?

Edit: of course any application of AC is equivalent to an application of the Zorn's Lemma. But I am sure you agree that some proofs work better with AC, some work better with the Well ordering axiom and some work better with Zorn's Lemma. When we prove that any vector space has a base we use Zorn's lemma. Nobody writes a proof with well ordering axiom while anybody can write such a proof.

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Why do people vote up questions like this? –  S. Carnahan Jan 15 '13 at 10:53
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Jekteem, please give us an example of an application of the Axiom of Choice in number theory, and explain why it is not equivalently an application of Zorn's lemma. –  S. Carnahan Jan 15 '13 at 10:54
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Hindman's theorem has a proof using Zorn's lemma. See, e.e., here: math.toronto.edu/lgoldmak/Hindman.pdf –  Péter Komjáth Jan 15 '13 at 12:41
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And I suppose existence of maximal ideals does not count as "number theory"? –  Andrej Bauer Jan 15 '13 at 12:56
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Concerning the last sentence of the question and Emil's comment: I sometimes find it helpful to use transfinite induction instead of Zorn's Lemma because it makes it easier to think (and write) about interleaving the construction with new sorts of steps, in order to obtain some additional properties of the object being constructed. –  Andreas Blass Jan 15 '13 at 13:52
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closed as not a real question by Simon Thomas, Tom Leinster, Felipe Voloch, Dan Petersen, Bruce Westbury Jan 15 '13 at 18:29

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1 Answer

There is a metatheorem that the Axiom of Choice is not necessary to prove any statements of Arithmetic, any proof of a statement about integers that uses AC can be constructively transformed into a proof which does not use that axiom. This comes under the heading of Shoenfield Absoluteness; see this Wikipedia article, and also this MO answer.

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A precise statement or reference would be nice here. –  Todd Trimble Jan 15 '13 at 15:07
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Todd: Shoenfield Absoluteness. See, e.g., this MO question which contains a short summary of the result: mathoverflow.net/questions/71965/… –  Grant Olney Passmore Jan 15 '13 at 15:23
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However, number theory is usually understood to encompass much more than arithmetic: MSC Tools from real and complex analysis are often used, which suggests that countable choice or DC may play a role. On the other hand, proofs from number theory often have a very "effective" taste. –  Goldstern Jan 15 '13 at 15:30
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Thanks, Grant. I'm going to edit that remark into the answer, under the assumption this is what Joe Shipman had in mind. If it's not, then he can re-edit to indicate what he did have in mind. –  Todd Trimble Jan 15 '13 at 15:50
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Just out of curiosity: Is there any relation between AC and Euclidlean Geometry ?? –  user30338 Jan 15 '13 at 17:21
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