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Excellent question. 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

Excellent question. 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.

dTal 6y ago
I don't understand Banach-Tarski, but it seems like the common version which you just explained must be wrong. It is not possible to double the volume of an object by cutting and reassembling. So someone has either made a mistake somewhere, or the "paradox" is using highly perverse definitions of "ball", "piece" etc such that if you understand what bizzare and unphysical concept was represented by them, the surprise goes away. The counterintuitive nature of the popular version is simply a lie, one way or the other.
- IngoBlechschmid 6y ago
  You are spot on! The paradox uses a, from the point of view of everyday life, highly perverse definition of "piece". 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.
- tsimionescu 6y ago
  The core of the matter is in the word "cutting". It is possible to pick points off of a connected sphere in such a way that you have 5 different collections of points on the surface of the original sphere (they are NOT connected surfaces). You then take these 5 sets, apply rotations and translations, and at the end, you have 2 spheres of identical volume. During the process, no 2 points from the same piece have changed their relative positions (of course, points from different pieces have). 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.
- mcherm 6y ago
  > 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 https://www.irregularwebcomic.net/2339.html
  - xyzzyz 6y ago
    > 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. 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.
- atomack 6y ago
  One point regarding whether or not this is unphysical is that in physical systems, conceptually we work in finite systems and at the only very end of our calculations we take the infinite limit while holding physically observable quantities fixed. This is the essence of the thermodynamic limit. 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

dTal 6y ago

I don't understand Banach-Tarski, but it seems like the common version which you just explained must be wrong. It is not possible to double the volume of an object by cutting and reassembling. So someone has either made a mistake somewhere, or the "paradox" is using highly perverse definitions of "ball", "piece" etc such that if you understand what bizzare and unphysical concept was represented by them, the surprise goes away. The counterintuitive nature of the popular version is simply a lie, one way or the other.

IngoBlechschmid 6y ago
You are spot on! The paradox uses a, from the point of view of everyday life, highly perverse definition of "piece". 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.
tsimionescu 6y ago
The core of the matter is in the word "cutting". It is possible to pick points off of a connected sphere in such a way that you have 5 different collections of points on the surface of the original sphere (they are NOT connected surfaces). You then take these 5 sets, apply rotations and translations, and at the end, you have 2 spheres of identical volume. During the process, no 2 points from the same piece have changed their relative positions (of course, points from different pieces have). 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.
mcherm 6y ago
> 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 https://www.irregularwebcomic.net/2339.html
- xyzzyz 6y ago
  > 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. 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.
atomack 6y ago
One point regarding whether or not this is unphysical is that in physical systems, conceptually we work in finite systems and at the only very end of our calculations we take the infinite limit while holding physically observable quantities fixed. This is the essence of the thermodynamic limit. 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