What is the Axiom of Choice?
My favourite statement of the Axiom of Choice, from AlienAtSystem via <a href="https://www.irregularwebcomic.net/4035.html" rel="nofollow">https://www.irregularwebcomic.net/4035.html</a> :<p>> As there are infinitely many sets, instead of going through them manually, you have to find an algorithm [to select an item from each set]. So Person A tells the other player an algorithm to select an element. Then that other player finds a set where the algorithm fails.<p>> If you, for example say: "Take the smallest element", then I say "One of my sets is all negative integers, it doesn't have a smallest element."<p>> Then maybe you say: "Take the smallest element, or if it does not have that, take the largest." Then I say "One of my sets is all integers, which has neither". Then maybe you add: "If it has neither, take the element closest to 0 (in case of tie, take the larger one)". Then I say: "What about the set of all integers and all their reciprocals? It doesn't have an element closest to 0 either."<p>> When Person A gives up and says: "Look, there's <i>some</i> algorithm that does it, right, why do I have to recite it manually?", then that person has taken the Axiom of Choice. If they say "Okay, fine, you win, there's no algorithm you can't find a counterexample to", then that person rejects the Axiom of Choice.