Re: I don't like the Axiom of Choice

On Apr 30, 1:00 pm, "Dave L. Renfro" <renfr...@xxxxxxxxx> wrote:
Craig Feinstein wrote (in part):

In other words, how much of modern mathematics, particularly
number theory and discrete mathematics, is based on the axiom
of choice? (I understand that Wiles' proof of FLT is based
on the Axiom of Choice. Am I correct? If so, then I guess
FLT is still an open problem, according to my reckoning!)

Mike Kelly wrote (in part):

No idea. It's probably worth bearing in mind that many,
many proofs that do rely on Choice can be reformulated
in a way such that they do not. Oftentimes, Choice is
just a convenience.

Unless you're more careful about exactly which forms
of the axiom of choice you consider suspect, I think
it's silly to even be worrying about the axiom of choice.
Each of the following CAN BE FALSE in the absence
of the axiom of choice:

1. The equivalence of the sequence definition of continuity
at a point with the epsilon-delta definition of continuity
at a point.

2. The reals are not a countable union of countable sets.

3. A countable union of measure zero sets has measure zero.

4. A countable union of first category sets is a first
category set.

5. Every infinite set has a countable subset.

There are many other results which virtually no one would
dispute that could be added to this list. Gregory Moore's
book on the axiom of choice is a great source for
these things. See also the following:

http://www.emis.de/journals/CMUC/pdf/cmuc9703/herrli.pdf

This post was absolutely fascinating and I'd like to read more about
this. Is this the book you are talking about?

http://www.amazon.com/Zermelos-axiom-choice-development-mathematics/dp/0387906703/ref=sr_1_2/103-3637349-8185408?ie=UTF8&s=books&qid=1177963872&sr=1-2

Also I am having a hard time with your link.

Thanks
Jason

.

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