>>5697214

>In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that "the product of a collection of non-empty sets is non-empty"

I consider this formulation polemic.

You have a bunch of other definitions in the background together with a large universe, this is why this happens.

The construction of the product is a set (per definition, so it IS a set in any case) whose elements are such and such. In your set theory without choice, "the product of non-empty sets results to be an empty set" is true, but only because the defintion requires you to collect the elements the set is supposed to contain, and then, without the axiom of choice, you can't find any. Hence the definition returns the empty set.

To say

"In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that 'the product of a collection of non-empty sets is non-empty' "

subliminally suggests that if you negate the axiom, then the theory is more well behaved - yes, but only because you robbed it it's power.

As far as opionions go, I' currently in a melancholy mood, telling me to not care too much about the man made desire to have a single framework for foundation.

That is I learn towards computation and complexity considerations now.