> the axiom of choice requires that you have a countable number of sets that you are choosing elements from You're referring to the "axiom of countable choice"

> the axiom of choice requires that you have a countable number of sets that you are choosing elements from You're referring to the "axiom of countable choice", which is a different axiom. The Wikipedia entry for the axiom of choice makes it clear that the number of sets can be uncountable. > select the sequence in which the "finite prefix" is all zeros This doesn't really make sense as a selection rule. The size of the prefix can vary between pairs of members from the same equivalence class.