I was reading some old threads and there was one (I don't have the link on hand, I can probably try to find it) where N was arguing that the AoC is illegitimately applied by measure theory and other stuff when dealing with the reals. He didn't really spell out the details behind his argument, so I'm not really sure how to interpret it. As far as I understand it, the axiom of choice would allow one to, for example, gather a set of real numbers that are unique up to some equivalence relation (so the equivalence classes are the "bins"). Certainly we have no reason to believe that these elements are all rational numbers... nor that the minimal extension containing a particular element will contain the others. That's pretty obvious. But what is the problem with this? I'm completely lost on this whole topic and I have searched google for some related information and I have found nothing to do with anything N was saying.

As far as I can recall, N failed to elaborate on his arguments on this subject, so I wouldn't spend too much time studying the discussion. If you disagree with something in mathematics, shouldn't you prove it wrong, or come up with a counterexample?Basically, in measure theory, you apply the AoC when you show that the Lebesgue measure of a countable disjoint union of measurable sets is equal to the measure of this union. This causes some strange results, as Banach and Tarski have shown: Banach-Tarski paradox. Thus some prefer to reject the AoC like N, and others to merely think of this as strange results when dealing with large sets.

I think the Axiom of Choice is hardly ever needed in 'any' real life. If you are doing a PhD on the foundations of mathematics you might need it but not otherwise. Don't bother. kili,What is your interest in AOC? QuoteDedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom).

QuoteOriginally posted by: CuchulainnI think the Axiom of Choice is hardly ever needed in 'any' real life. If you are doing a PhD on the foundations of mathematics you might need it but not otherwise. Don't bother. kili,What is your interest in AOC? QuoteDedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom). I have just been thinking a bit about the philosophy of math (or at least the philosophy of applied math). The notion of a proof in math differs from the scientific method, in that we are dealing with a formal system in which there really is concrete notion of truth (except unfortunately, it has no particular bearing on reality). As Stale mentioned, there are "paradoxes" when you make strong enough assumptions. Really the notion of paradox such as Banach-Tarski is just that the results are counterintuitive... people have assumptions about how the universe works and they project these onto the mathematics. But in reality there are no sets that large (to our knowledge) so it's sort of like arguing that all unicorns are purple.It works both ways too; people often look at the mathematics and "prove" some fact about reality. In truth they are proving some conditional proposition -- "if the axioms we hold to be true are consistent with the universe, then ...", but they seem to ignore the assumptions made and just assume the results. An example of this would be the notion of black holes as singularities. As far as I am aware, there is no reason to believe that those sort of singularities exist in the real world ... only that they are the result of some infinite limiting process (which may or may not be bounded in time). Similarly, it is argued by many that QM is a method for predicting results, not explaining them... this has been beat dead in many random vs deterministic arguments. But when you don't understand all of the details it is easy to be misled by pop physics books explaining "how" the universe operates.But the AOC thing in particular was just curiosity on my part; the statement about field extensions somehow invalidating the AOC was pretty enigmatic.

On my part, I don't loose any sleep trying to justify the use of the AoC. I quote Folland, from his classic "Real Analysis" out on John Wiley & Sons;QuoteThe AoC is generally taken as one of the basic postulates in the axiomatic formulations of set theory. Some mathematicians of the intuitionist or constructivist persuation reject it on the grounds that one has not proved the existence if a mathematical object until one has shown how to construct it in some resonable explicit fashion, whereas the whole point of the AoC is to provide existence theorems when constructive methods fail (or are too cumbersome for comfort),and he makes use of the theorem on several occations.What's good enough for Folland, will certainly be good enough for me