choices

There were three profound topics that I recall debating in my first year as a graduate student.
I mean real student debates over a heterogenous assortment of alcohol and gallons of bad coffee.

One was whether the Clash were sell-outs or deeply sardonic (sold-out, clearly);
one was on the pros and cons of unionization (long story, MIT libertarians vs Europeans);
but the longest was on the Axiom of Choice

Mark C-C explains the axiom of choice

I like the "given a collection of bins each containing at least one object, exactly one object from each bin can be picked and gathered in another bin" - because it is trivially and obviously true for finite sets, even to a physicist.

But the real import of the AoC is for infinite sets, and the "choice function" definition was the one we wheeled around.

I started off in the constructivist camp, and was deeply suspicious that it was presented as an axiom, it felt like it ought to be a theorem - derivably true from the more basic axioms.
I don't know how many hours it took James to patiently wear me down (and I don't think I ever convinced him in turn that Strummer really had sold out), but I think the Alepth invocation of "by the Axiom of Choice we can..." in response to one of my counter-points was sufficient to make me understand that it really is an axiom.

But... I think Mark makes one slightly misleading comment, when he says "...in the end, Cantor and his successors won out. Now the axiom of choice is pretty much universally acknowledged as valid..."

See, is is an axiom.
So we can choose its negative as an axiom, if we want.
That for some suitably infinite set there is not a choice function, and that we can not rank the cardinality of some infinite sets, or that there are infinite vector spaces for which a basis set can not be constructed.

That is interesting, because the axiom of choice is useful, but need not necessarily be true.

I also felt guilty when I noticed I'd bumped Mark off the Physical Science headers.