To give you guys the context: This isn’t for anything specifically, however, I hear the phrase such and such “…then we invoke the axiom of choice…” or something like, “…but by the axiom of choice, we derive Zorn’s lemma” or anything along those lines.
My question: How did the axiom of choice have anything to do with what they are saying? Literally, the only time the axiom of choice is ever invoked in my life has been only at times where I do not see how it would be implied. Intuitively, I understand what the the basic descriptions are, but I logically cannot follow the axiom of choice whenever a professor or a professional ever uses the term.
I get there is some subtle point I’m missing, but I’m trying to understand how the axiom of choice is ever implied.
Let me phrase it this way: If I was talking to someone, and I was explaining the logic behind something, when do I tell the other person that I am invoking the axiom of choice?