Irregular Webcomic

My favourite statement of the Axiom of Choice, from AlienAtSystem via https://www.irregularwebcomic.net/4035.html :

> 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.

> 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."

> 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."

> When Person A gives up and says: "Look, there's some 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.

> I don't understand Banach-Tarski, but it seems like the common version which you just explaing must be wrong. It is not possible to double the volume of an object by cutting and reassembling.

Well, it seems that you DO understand why it is considered to be very surprising and unintuitive.

The definitions of "ball" and "rotate" are perfectly normal. The definition of "piece" is slightly odd in that the "pieces" don't have smooth surfaces -- they are "jagged" or "fuzzy" down to an infinite level. (This is why common notions like "volume is conserved" don't apply -- such notions don't apply normally even to simpler objects like a fractal Serpinski Sponge.)

What is especially weird about the pieces (axiom-of-choice weird) is that specifying exactly what the boundaries of the pieces are is so hard that no clear description or algorithm can be given that specifies them. It's sort of like if you said "cut a ball so the prime-numbered points are in piece 1 and the composite points are in piece 2", except MORE strange because the definition of "prime" is easy to understand.

My favorite explanation is this one: https://www.irregularwebcomic.net/2339.html

Actually, reading the comic another poster put up here:

http://www.irregularwebcomic.net/2339.html

... I think I've managed to get my head round it. Or, at least form an initial model.

It's like the old puzzle of the hotel with infinite rooms, each one labelled with an integer.

Then, another infinite set of guests turn up. The hotel manager asks every person currently in the hotel to double their room number and go to that one. Because there are infinite even numbers, this works fine.

Then, the new set of infinite guests sleep in the odd number rooms.

If I'm right, of course, then this only works because infinity is really odd - and you couldn't get an infinity room hotel in the real world.

Is this a reasonable way of thinking about it?

This article (linked elsewhere on this page) is a good description: http://www.irregularwebcomic.net/2339.html

Once I understood this, I find it is a bit a cheat (as you might expect). The 'cuts' are not cuts in any real-world sense.

Hopefully you are happy with the idea that there are as many integers as there are both even integers or odd integers. Therefore in some sense I can "cut" the integers into two equally sized sets. This paradox does something very similar on the real numbers, making use of the form of the real numbers to avoid the "doubling" effect I got when cutting the integers.