What is the Axiom of Choice?
My favourite statement of the Axiom of Choice, from AlienAtSystem via <a href="https:&#x2F;&#x2F;www.irregularwebcomic.net&#x2F;4035.html" rel="nofollow">https:&#x2F;&#x2F;www.irregularwebcomic.net&#x2F;4035.html</a> :<p>&gt; 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>&gt; If you, for example say: &quot;Take the smallest element&quot;, then I say &quot;One of my sets is all negative integers, it doesn&#x27;t have a smallest element.&quot;<p>&gt; Then maybe you say: &quot;Take the smallest element, or if it does not have that, take the largest.&quot; Then I say &quot;One of my sets is all integers, which has neither&quot;. Then maybe you add: &quot;If it has neither, take the element closest to 0 (in case of tie, take the larger one)&quot;. Then I say: &quot;What about the set of all integers and all their reciprocals? It doesn&#x27;t have an element closest to 0 either.&quot;<p>&gt; When Person A gives up and says: &quot;Look, there&#x27;s <i>some</i> algorithm that does it, right, why do I have to recite it manually?&quot;, then that person has taken the Axiom of Choice. If they say &quot;Okay, fine, you win, there&#x27;s no algorithm you can&#x27;t find a counterexample to&quot;, then that person rejects the Axiom of Choice.