Re: Have I understood the axiom of choice?

On Sat, 13 Sep 2008 06:42:57 -0700 (PDT), ALiX <alix.tofigh@xxxxxxxxx>
wrote:

I need to verify my current understanding of (the need for) the axiom
of choice. Let us work in ZF. Let S be a function on I, where I can be
any infinite set. It is easy to prove the existence of the set F of
all functions f on I such that f(i) in S(i) for all i in I. My
understanding of the axiom of choice is that it is equivalent to the
statement that F is non-empty.

Yes, that's exactly right.

(Well, what you _meant_ is exactly right. Of course you need
to _assume_ that S(i) is nonempty for all i in I.)

nformally, although we know that the
set of all "sequences" on an indexed family of sets exists, we cannot,
in general, prove that this set is not empty without the axiom of
choice.

I also remember reading somewhere (perhaps in a newsgroup like this
one) that its possible to trace the whole issue to the distributivity
of 'and' over 'or', or something to that effect. I can't seem to find
that reference anywhere. Does this ring a bell with anyone?

Cheers,
/ALiX

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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