Question

Bertrand Russell pointed out that some applications of the axiom of choice are easier
to avoid than others. For instance, given an infinite collection of pairs of shoes,
describe a way of choosing one shoe from each pair. Could you do the same for an
infinite set of pairs of socks?

Answer 1

In case of shoes each pair of shoes comes with left and right type. It is identifiable.

Given an infinite collection of pairs of socks instead of shoes , there is no way to actually exhibit a "simultaneous" choice of a sock for each pair ofcourse. It is not like we have been literally given the same identical pair infinitely (uncountable) many times (if so, then there is a way: We make one sort of decision, and it can be copied it on each pair).

Indeed, for the pair of sock s, the ?n{0,1}?n{0,1} has size RR without any use of particular choice, even if (in case of consistently) some ?nAn?nAn with each |An|=2|An|=2 are again empty.