Re: Questions about Axiom of Choice


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?

It is now. Until relatively recently, an existence proof was required
to give a method of construction.

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

My favorite is the Banach-Tarski paradox: it is possible to take
the unit ball (i.e. radius 1) in three dimensional euclidean space,
divide it into a finite number of pieces, move those pieces around
via rigid motions (i.e. translations and rotations), and re-assemble
them into a ball of radius 2.

--Dan Grubb
.

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