Re: Questions about Axiom of Choice

In article <1157467811.267002.47890@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<agapito6314@xxxxxxx> wrote:
Hi, I have seen 2 types of objections by those who question the AC:

1.- It is a "pure existence" statement, that is it tells something
exists but not what it is. Is math not full of these kinds of
statements?

Aatu has given a far better answer to this than I could ever hope to...


2.- Some of its implications are weird or counterintuitive. Can
someone please explain some of these?

Counterintuitive depends on your intuition... But among two "typical"
examples:

(i) The well-ordering principle; in particular, the conclusion that
the real numbers can be well ordered.

(ii) The Banach-Tarski Paradox (which is also a theorem): there is a
way to partition the unit sphere into a finite number of parts,
and then to take those parts, and reassemble them (without
changing their shape or even reflecting them) into two unit
spheres (or into a sphere of radius 2).


--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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