Suppose there is a well-known theorem whose usual proof uses Axiom of Choice. Is trying to prove it without Axiom of Choice useless? What merits can such a proof have?
Is trying to prove a theorem without Axiom of Choice useless?
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5http://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice/ – 2012-07-15
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18This is an odd question to ask, after you have posted 8 questions about proving one theorem or another without the Axiom of Choice! Shouldn't you have decided whether it was useful first, before asking all those questions? – 2012-07-15
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0@GerryMyerson As the comments to my questions show, not so small number of members don't seem to understand the merits of such proofs. So I thought this thread might help them understand. – 2012-07-15
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1@GerryMyerson Even if I know an answer, there can be other(or even many) answers that I don't know. – 2012-07-15
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0@QiaochuYuan I have no idea why my answers were deleted. Please explain to me as I don't want this kind of thing to happen again. – 2012-07-15
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17@Makoto: I think it is disingenuous to post a question like this and immediately post a response defending your other questions on the site. This kind of soapboxing is what blogs are for, not math.SE. – 2012-07-16
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0@QiaochuYuan Ansering one's own question has no problem in this site. I have no idea why it is disingenuous. – 2012-07-16
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0I'd like someone to open a meta thread on this subject. I don't have a right to open it. – 2012-07-16
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0Dear Makoto: Do you know the book *Théorie des ensembles* (translated as *Theory of Sets*) by Nicolas Bourbaki? – 2012-07-16
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3If you would like to open a thread on meta you can *register* and thus gain quite a lot of privileges. Amongst them is voting and posting on meta. – 2012-07-16
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0@Pierre-Yves Yes, but I've never read it. I have most of their volumes(including theory of sets), so I can check it. – 2012-07-16
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0@AsafKaragila What are the pros and cons of registering? – 2012-07-16
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1Dear Makoto: From Bourbaki's perspective, the expression "axiom of choice" just doesn't make sense. Of course, most of the mathematicians don't use Bourbaki's theory, but there are important exceptions, like Serre and Grothendieck (and many others). I just wanted to make sure you were aware of this. – 2012-07-16
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0The pros are having all sort of privileges on this site, the only con I can think of is that you are required to supply some email address (which is visible only to moderators, though), but I think one can overcome this as well. – 2012-07-16
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0@Pierre-Yves That's interesting. I'll check it. Thanks. – 2012-07-16
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0@AsafKaragila Thanks. I'll consider it. – 2012-07-16
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0@Pierre: I have not read Bourbaki's book, but I have heard that they in fact do not use ZF at all, but rather the much weaker Z[ermelo] set theory. Is that true? – 2012-07-16
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0@QiaochuYuan I can write my deleted answers in my questions concerning AC instead of writing the link to this question. I believe it's perfectly okay. The end effect would be the same. May I ask again what's wrong with my deleted answers? – 2012-07-16
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0Dear @Asaf: I don't know. Sorry. Here are four links which seem in tune with what you heard: http://mathoverflow.net/questions/14356 --- http://personnel.univ-reunion.fr/ardm --- http://www.rbjones.com/rbjpub/logic/jrh0105.htm --- http://jfr.unibo.it/article/view/1899 – 2012-07-17
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0@Pierre: In this case, I should probably "counter-link" the offer to read Bourbaki with Mathias paper "*[The ignorance of Bourbaki](http://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf)*". – 2012-07-17
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0@QiaochuYuan "Publishing" a mathematical idea in a form of Q&A is being encouraged in this site. "Since Stack Overflow launched, we've been trying to explain that it's not just a Q&A platform: it's also a place where you can publish things that you've learned: recipes, FAQs, HOWTOs, walkthroughs, and even bits of product documentation, as long you format it as a question and answer." http://blog.stackoverflow.com/2012/05/encyclopedia-stack-exchange/ – 2012-07-18
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0@QiaochuYuan For fairness' sake, I'd like to inform you that I opened a meta thread on this subject. http://meta.math.stackexchange.com/questions/4677/in-what-case-answering-ones-own-question-should-be-forbidden – 2012-07-18