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I do not really understand why the Banach-Tarski paradox is such a big deal. It does not seem any different from the paradoxes of the Hilbert hotel. If one has

by cjfd 6y ago

I do not really understand why the Banach-Tarski paradox is such a big deal. It does not seem any different from the paradoxes of the Hilbert hotel. If one has two filled hilbert hotels one can fit all hotel guests in one single hilbert hotel by letting guests from one hotel take even numbered rooms and from the other hotel odd numbered rooms.

magnio 6y ago
They are both counter-intuitive. The difference is that proving Banach–Tarski paradox requires the axiom of choice.
- cjfd 6y ago
  Yes, but I think the basic cause of both is that one assumes there are infinite sets. The Banach-Tarski paradox to me seem to be little more than a three dimensional variation of the hilbert hotel. Therefore, if one wants to implicate any basic axiom for being the cause of the counter-intuitiveness of that theorem the infinity axiom seems the more sensible candidate. Therefore, to me, it seems the argument against the axiom of choice from the Banach-Tarski paradox does not make any sense.
  - ukj 6y ago
    You are disagreeing over the meaning of the word "the" in "THE axiom of choice". It's not only one - there are multiple formulations. There are the axioms of finite choice; and the axiom of infinite choice. Broadly it's the philosophical distinction between Finitists and Infinitists.
    - cjfd 6y ago
      I don't really think so. I am arguing that whatever axiom of choice one uses is basically an implementation detail of the banach-tarski paradox and that because of the similarity to the hilbert hotel paradox the cause of the paradox should be sought in the infinity axiom and not in whatever axiom of choice one prefers or doesn't prefer. Edit: or to put it more sharply: after one has seen the hilbert hotel the BT paradox is not surprising anymore.
      - ukj 6y ago
        The finite axiom of choice doesn't suffer from paradoxes of infinity. 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/Nondeterministic_programming https://en.wikipedia.org/wiki/Nondeterministic_programming
        cjfd 6y ago
        Well, if one assumes finiteness all of these paradoxes go away and this discussion is not about anything anymore. I studied physics and am quite attached to infinities. That one constrains oneself to a finite state system mostly does not mean that one is not allowed to look out of the window towards the stars and marvel at infinity anymore.
        ukj 6y ago
        From your comment above I am making the (admittedly presumptuous) inference that you didn't study Quantum physics. https://en.wikipedia.org/wiki/Renormalization 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.
        cjfd 6y ago
        Well, I did a PhD in quantum field theory and as part of that published a paper on renormalization..... Of course the infinite sets involved can be approximated using finite sets if one does not mind losing important symmetries. Still, these finite sets would still be frightingly and incomprehensibly large.... If one wants to comprehend anything one may be better off just ignoring quantum mechanics..... And I am not talking about the measurement problem. I am talking about the incredible vastness of the state space. The wave function of the universe is an incredible and completely untractable thing even if one assumes it is finite. It contains all possible histories of the universe for one thing and lets all of these possible histories interact with one another.
hugh-avherald 6y ago
They're quite different. The Banach-Tarski paradox requires a finite number of pieces and physically simple transformations. It's not simply the discovery of a bijection between a sphere and two spheres.
IngoBlechschmid 6y ago
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.
  - 6y ago

by cjfd 6y ago

magnio 6y ago

They are both counter-intuitive. The difference is that proving Banach–Tarski paradox requires the axiom of choice.

cjfd 6y ago
Yes, but I think the basic cause of both is that one assumes there are infinite sets. The Banach-Tarski paradox to me seem to be little more than a three dimensional variation of the hilbert hotel. Therefore, if one wants to implicate any basic axiom for being the cause of the counter-intuitiveness of that theorem the infinity axiom seems the more sensible candidate. Therefore, to me, it seems the argument against the axiom of choice from the Banach-Tarski paradox does not make any sense.
- ukj 6y ago
  You are disagreeing over the meaning of the word "the" in "THE axiom of choice". It's not only one - there are multiple formulations. There are the axioms of finite choice; and the axiom of infinite choice. Broadly it's the philosophical distinction between Finitists and Infinitists.
  - cjfd 6y ago
    I don't really think so. I am arguing that whatever axiom of choice one uses is basically an implementation detail of the banach-tarski paradox and that because of the similarity to the hilbert hotel paradox the cause of the paradox should be sought in the infinity axiom and not in whatever axiom of choice one prefers or doesn't prefer. Edit: or to put it more sharply: after one has seen the hilbert hotel the BT paradox is not surprising anymore.
    - ukj 6y ago
      The finite axiom of choice doesn't suffer from paradoxes of infinity. 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/Nondeterministic_programming https://en.wikipedia.org/wiki/Nondeterministic_programming
      - cjfd 6y ago
        Well, if one assumes finiteness all of these paradoxes go away and this discussion is not about anything anymore. I studied physics and am quite attached to infinities. That one constrains oneself to a finite state system mostly does not mean that one is not allowed to look out of the window towards the stars and marvel at infinity anymore.
        ukj 6y ago
        From your comment above I am making the (admittedly presumptuous) inference that you didn't study Quantum physics. https://en.wikipedia.org/wiki/Renormalization 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.
        cjfd 6y ago
        Well, I did a PhD in quantum field theory and as part of that published a paper on renormalization..... Of course the infinite sets involved can be approximated using finite sets if one does not mind losing important symmetries. Still, these finite sets would still be frightingly and incomprehensibly large.... If one wants to comprehend anything one may be better off just ignoring quantum mechanics..... And I am not talking about the measurement problem. I am talking about the incredible vastness of the state space. The wave function of the universe is an incredible and completely untractable thing even if one assumes it is finite. It contains all possible histories of the universe for one thing and lets all of these possible histories interact with one another.

hugh-avherald 6y ago

They're quite different. The Banach-Tarski paradox requires a finite number of pieces and physically simple transformations. It's not simply the discovery of a bijection between a sphere and two spheres.

IngoBlechschmid 6y ago

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.
- 6y ago