The Point of the Banach-Tarski Theorem

> "And that's how you can make an infinite amount of space in your infinite-but-full hotel."

If you re-frame that as an inexhaustible resource than you can make any amount of space in it because by definition it is inexhaustible.

> "infinite-but-full..."

"If it were already full then..." It can't be full. By definition it is inexhaustible.

"but we have an infinite supply of hotel guests." So by definition we can't house them all because the supply cannot be exhausted.

"We fill this inexhaustible space with an inexhaustible quantity of stack-able objects..." It seems as nuts as saying "We apply this irresistible force to this immovable object.." The existence of one implies the other _cannot_ exist.

Pick one, reason with it. Include a second in your reasoning and... No.

A one to one mapping between two resources that we define to be inexhaustible boils down to take infinity and divide it by infinity. How can it be anything else? So yes you can get 2, or 10, or pi, or literally anything at all. Infinity - infinity is no better, division is repeated subtraction so you still get "Gerald" as the answer.

Sure two infinite things are, by some view of it, "The same size" where that size makes them not comparable. The set of points contained in a sphere in my hand has the "same size" as the set of points in the sphere we call the earth which is the same as the set of points in the universe. But so what? How does this tell us anything? We divided by infinity to define what a point is and there you have to stop or you get nonsense, even if it is convenient nonsense I would hesitate to build my house there. Maybe it doesn't make any sense to talk about size when it isn't finite. Maybe we can only really reason about it using limits for the same reason we note dividing by zero makes no sense in any way that is really useful?

In that case, I would say you're a finitist (which, as the name seems to suggest, is fine).

The set theory approach gives seemingly clear answers to all of these questions by talking about domains and ranges of functions, rather than talking about what is or isn't an "inexhaustible number". There are potentially different correspondences available, represented by different functions; some possible correspondences may follow the "if you select corresponding to itself in preference" pattern, while others don't. = and ≤ for cardinalities are defined using existence of certain functions between sets.

However, you don't have to believe that any infinite sets exist or that we should be allowed to quantify over them, or that we should attempt to define cardinality for infinite sets at all. Still, as Dana Scott said in a related context, "if you want more, you have to assume more".

If it helps, don't think about operations on infinite sets as "things that are true (or false)". Rather, think about it as "an abstract model with certain rules, that can produce interesting results if applied in careful ways."

Like √-1. Does that exist? Well, not really. But if we make up a pretend number i that we say is equal to √-1, we can perform interesting mathematical operations on it that can still produce useful results in the real world. But only if we follow a given set of rules carefully and consistently.

"All models are wrong, but some models are useful."

Infinity isn't real - at least, not in our universe. It certainly isn't a number. But it's a concept we can use to produce useful results sometimes, if we handle it carefully. But it's also a concept that can produce some weird results, even if we follow all the rules.