Re: I don't like the Axiom of Choice

In article <1177948109.585123.246650@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Craig Feinstein <cafeinst@xxxxxxx> wrote:
I don't like the Axiom of Choice. It may be correct or it may be
incorrect, but I don't think it is anyone's business to assume claims
about the universe of transfinite sets without actually clearly
demonstrating that the claims are true. After all, lots of crazy stuff
happens when one deals with infinite sets. (My favorite example is
Hilbert's Hotel Infinity. http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel)
And this is what the Axiom of Choice is doing - making claims without
explicitly demonstrating why the claims must be true.

These transfinite "crazy items" have nothing to do with
the axiom of choice. The Hilbert Hotel example does not
in any way depend on it.

Throughout most of my mathematics education, the axiom of choice has
been assumed implicitly by my mathematics professors (or explicitly
without making a big deal about it). How much of my mathematics
education should I forget about?

Or more specifically, since I never really took geometry, analysis,
topology seriously because of my natural aversion to infinite sets,
how much of my number theory and discrete mathematics education should
I forget about?

Even number theory involves an infinite set, namely,
the set of integers. Euclid includes many results
about infinite sets of integers.

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!)

All of analysis involves at least the real numbers.

I am not at all familiar with the proof of FLT, but
I doubt it requires the axiom of choice. It does
use the existence of infinite sets.

Craig


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

Relevant Pages

  • Re: Why does Cantor a target for cranks?
    ... It has a lot to do with the choice also to do mathematics ... the two nonconstructive ones are the axiom of ... computable reals are complete and uncountable. ... and refrain from using infinite sets. ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... > which is not smaller han the cardinality of that set. ... all sets of naturals begs the very question of whether there are ... because we dobt the existence of infinite sets. ... >>> There is neither any axiom asserting the existence of actual ...
    (sci.math)
  • Re: Cantors circular "proof" that evens = integers
    ... And I am not saying he should not have been talking: ... If it helps to make the process shorter, I already know that if the axioms of set theory can be used to prove that "2n e N whenever n e N," then it follows that N ~ E. In Cantor's day, since no one was even dealing with infinite sets, obviously no one had answered questions such as, "in an infinite set, can we assume that if n e N, then 2n e N? ... What I found interesting about Moe's post was that 2n e N if we can use the axiom of choice, but you're right, I missed or forgot that you said that this is true under the axiom of choice. ... However, remember that you gave an unsupported CLAIM, and until I see the proofs that it follows from the axiom of choice that N ~ E, I cannot see whether such proofs do the same thing that the proofs I have so far seen that N ~ E do, namely ASSUME, somewhere, something equivalent to 2n e N. For what it's worth, I am certain that 2n is NOT e N is a perfectly valid alternative axiom -- again, see the Dichotomy -- and that we have either two valid branches of mathematics, like Euclidean and non-Euclidean geometry, or we have ONE valid branch, with the 2n e N being the branch that gets tossed out. ...
    (sci.logic)
  • Re: Galileos Paradox and the Project of the Reals
    ... Please state the axiom of infinity. ... Set theory doesn't talk about "size" as far as I ... Cardinality is an extension of finite size, ... So, when it comes to infinite sets, this notion of order, however it is implemented in the definition, should be considered in relative "set" measures. ...
    (sci.math)