The Hilbert's-Hotel-style result that the union of two balls consists of exactly the same amount of points than just a single ball is much more basic than the Banach–Tarski paradox.
In the Banach–Tarski paradox, we cut a single ball into a finite number of pieces (incidentally, it can be done with just five). These pieces are then rotated and moved in space, but otherwise kept exactly as they are. The surprising fact is that after moving and rotation, the five pieces fit together to form two balls.
The definition of "piece" used is: "Arbitrary collection of points which need to bear any relation." This is in contrast to pieces of things in everyday life, which are firstly not made of infinitesimal points, but even if we keep rolling with that, satisfy the restriction that with every included point also some neighboring points are included.
This is not so with the "pieces" appearing in the Banach–Tarski paradox. In fact, they are so extremely rigged/fractal/dislocated/non-contiguous, that it is not possible to assign a meaningful measure of volume to these. For instance, the unit cube has volume 1. The empty set of points has volume 0, and so do sets which contain just a finite number of infinitesimal points. But the "pieces" appearing in the paradox are so weird that we cannot meaningfully attach any volume at all to them.
Also, there can never be any formula or any other explicit description of which points the five pieces consist of. This is because the proof employs the axiom of choice.
The 5 pieces are highly un-intuitive objects. I imagine if you were to realize even an approximation of them, they would each look like a complete sphere, with infinitely many infinitesimally small holes poked through. But when you do put them together, each and every one of the (uncountably) infinitely many holes in the 1 sphere is perfectly plugged by one of the (uncountably) infinitely many points in one of the other spheres.
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
No, volume always is conserved just fine, as long as you deal with measurable pieces. Sierpinski’s sponge has a well defined volume, and this volume behaves in a sensible manner. Normal sets have well defined volume.
The thing with Banach-Tarski pieces is that they cannot be assigned any volume in any sensible way. It has nothing to do with them being “jagged” or “fuzzy”, but rather with their weird behavior when it comes to self-overlapping translations.
In these paradoxes, infinities are present from the outset and I think it's this that leads to the unphysical outcome. They're not wrong. They are mathematical paradoxes. But it's not a problem they are unphysical (from a physics point of view) because physics uses mechanisms, like the thermodynamic limit, to handle infinite limits sensitively. Then the paradox goes away.
For instance, the 'physics' version of the Hilbert hotel problem would say there are two hotels, one with N rooms and the other with M rooms. Then do all the renumbering you like, the paradoxical situation of filling both hotels and then putting all guests from both hotels into one of them is no longer possible. Finally, if you want to think about hotels with an infinite number of rooms take N and M to inifinity keeping N/M fixed
Edit: add physicified version of Hilbert hotel problem
There are the axioms of finite choice; and the axiom of infinite choice.
Broadly it's the philosophical distinction between Finitists and Infinitists.
Edit: or to put it more sharply: after one has seen the hilbert hotel the BT paradox is not surprising anymore.
It's mostly intuitive to computer scientists that infinities don't exist ;)
And we are also in the habit of proving/realizing our choice-functions rather then assuming them axiomatically.
https://en.wikipedia.org/wiki/Renormalization
If giving up infinities allows me to understand The Universe better, I don't see how that robs me of marveling at the stars - it only makes it more exciting!
I can't comprehend an infinite universe. Nobody with finite memory/time can.
We can't comprehend the universe using numerical methods - it's too complex for our brains/computers.
But we can understand complexity using symbolic methods.
Symbolism/representationalism (religion) is rather inevitable part of the human condition.
∞
Basically, it doesn't hold for any constructive logic.
https://mathworld.wolfram.com/Banach-TarskiParadox.html
It is an existence proof that relies on the Axiom of Choice, of course there is no algorithm to actually construct those pieces.
This is one of the reasons that people distinguish between ZFC and ZF (Zermelo-Fraenkel with and without AC).
ZFC is not regarded as sound by some because of these Paradoxes.
Can you give relevant pointers? I'm curious because while I do know a couple of arguments for the inconsistency of ZFC, none of these are related to the Banach–Tarski paradox or similar nonintuitive results.
Also I'd like to stress that ZFC and ZF are exactly as consistent: If ZF is consistent, then so is ZFC (and vice versa). This meta-result is proven in a very weak finitary logical meta-system ("PRA"), hence can be trusted even if one is wary of set-theoretic infinities. The keyword is "Gödel's constructible hierarchy".
In this line, there are even more astonishing meta-theorems, all provable in PRA:
* From any ZFC-proof of a purely number-theoretical statement (a statement which refers only to natural numbers), any usage of the axiom of choice can be mechanically eliminated. That is, any ZFC-proof of such a result can be transformed into a ZF-proof.
* From any ZFC-proof of a statement of the form "for all numbers n, there exists a number m such that ...", where in "..." no more quantifiers ("for all", "there exist") appear, any usage of the law of excluded middle ("any statement is either true or not") can be mechanically eliminated. That is, any such ZFC-proof can be transformed into a proof in IZF ("Intuitionistic Zermelo–Fraenkel"). The keyword is "double negation translation".
These meta-theorems indicate that the axiom of choice and the law of excluded middle can be regarded as "mathematical phantoms". Just as the complex numbers work wonders for us but can be compiled away to pairs of real numbers, these set-theoretic principles promise to work wonders for us and their usage can be compiled away if we wish so.
B-T is Hilbert Hotel plus Axiom of Choice.
On the other hand, to me it sounds like the distinction may also come from the difference between countable infinities (which Hilbert's Grand Hotel is limited to), and uncountable infinities (which BT depends on).