The Banarch-Tarski Paradox

[Magical Realist]

The Banach–Tarski paradox is a theorem in set-theoretic geometry which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces (i.e., disjoint subsets), which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated colloquially as "a pea can be chopped up and reassembled into the Sun."

The reason the Banach–Tarski theorem is called a paradox is that it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations preserve the volume, but the volume is doubled in the end.

Unlike most theorems in geometry, this result depends in a critical way on the choice of axioms for set theory. It can be proven only by using the axiom of choice,[1] which allows for the construction of nonmeasurable sets, i.e., collections of points that do not have a volume in the ordinary sense and that for their construction would require performing an uncountably infinite number of choices."--http://en.wikipedia.org/wiki/Banach–Tarski_paradox

Can you do this with a real physical ball?

 

[mathman]

The Banach–Tarski paradox is a theorem in set-theoretic geometry which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces (i.e., disjoint subsets), which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated colloquially as "a pea can be chopped up and reassembled into the Sun."

The reason the Banach–Tarski theorem is called a paradox is that it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations preserve the volume, but the volume is doubled in the end.

Unlike most theorems in geometry, this result depends in a critical way on the choice of axioms for set theory. It can be proven only by using the axiom of choice,[1] which allows for the construction of nonmeasurable sets, i.e., collections of points that do not have a volume in the ordinary sense and that for their construction would require performing an uncountably infinite number of choices."--http://en.wikipedia.org/wiki/Banach–Tarski_paradox

Can you do this with a real physical ball?

No. Non-measurable sets are non-physical.

 

[chinglu]

No. Non-measurable sets are non-physical.

So how does this resolve the paradox?

Also, since non-measurable sets are sets, does this imply ZFC sets are not compatible with physical reality?

 

[Lakon]

So how does this resolve the paradox?

Good question.

 

[RJBeery]

So how does this resolve the paradox?

Also, since non-measurable sets are sets, does this imply ZFC sets are not compatible with physical reality?

I've never appreciated this as a paradox because it's playing with infinities. A small ball of infinite density can be rescaled to any arbitrary size of equal density, for example. The fact that they can claim no stretching is used is the trick, but I still don't get the "paradox"...

 

[mathman]

So how does this resolve the paradox?

Also, since non-measurable sets are sets, does this imply ZFC sets are not compatible with physical reality?

Mathematic entities are never physical. Anything coming out of ZFC is a mathematical construct.

When physicists develop their theories, they use mathematics as a tool. For example a physical ball is made up of atoms, etc. with a lot of empty space, but when considering it macroscopically, it is treated as homogenous.

 

[Lakon]

I've never appreciated this as a paradox because it's playing with infinities. A small ball of infinite density can be rescaled to any arbitrary size of equal density, for example. The fact that they can claim no stretching is used is the trick, but I still don't get the "paradox"...

Ah .. of course. I missed the infinity bit. Where infinities are involved, anything is possible, I guess.

 

[eram]

I've never appreciated this as a paradox because it's playing with infinities. A small ball of infinite density can be rescaled to any arbitrary size of equal density, for example. The fact that they can claim no stretching is used is the trick, but I still don't get the "paradox"...

Hmm, I thought that a physical interpretation of the Barnach-Tarski process is that the two new balls have half the density of the original ball.

 

[mathman]

Hmm, I thought that a physical interpretation of the Barnach-Tarski process is that the two new balls have half the density of the original ball.

There is no physical interpretation. Any mathematical theorem involving non-measurable sets cannot be given a physical interpretation.

 

[AlphaNumeric]

So how does this resolve the paradox?

It is not a paradox, it is just counter intuitive.

Also, since non-measurable sets are sets, does this imply ZFC sets are not compatible with physical reality?

Mathematical constructs do not need to reflect reality. And part of a mathematical construct not having an application in physics doesn't mean said construct has no application in physics. Its a subtle point you, chinglu, are too stupid to grasp but the role of mathematics within physics is to attempt to formalise in a conceptual manner the structure observed within physical reality and to then explore logical implications of such conceptualisations.

We know Euclidean geometry is not consistent with many aspects of gravitational mechanics but that doesn't mean geometry is inconsistent with reality or that we cannot apply Euclidean concepts elsewhere.

I know you're always itching to find any excuse to try to dismiss/deride mathematical concepts such as set theory or the use of Lorentz groups in relativistic mechanics but you are prevented from doing a proper evaluation of the role and application of such concepts by the fact you don't understand them and refuse to try to learn anything about them.

Hmm, I thought that a physical interpretation of the Barnach-Tarski process is that the two new balls have half the density of the original ball.

No, the mathematical formalisation of modelling density is precisely based on the notion of measurable sets. Density is a measure of how much stuff is contained within a region, the same Riemann or Lebesgue measured used in abstract set theory are used in quantifying density of matter, charge, probability etc found in physical models.

The 'paradox' relies on something called the Axiom of Choice, which is a statement about how to count, in some sense, pieces in particular infinite sets. Before you cut up the ball you can quantify its density (if we're going to give this a physical interpretation in a manner akin to your query) as the set of points making up the ball is non-pathological and measurable. The AoC lets you then split the ball up into pieces which cannot be measured, you cannot assign a density to a collection of points making up a 'piece' as, unlike how you really cut up an apple, the 'piece' is some incredibly complicated fuzz of points which do not form 'chunks'. When you move these hazes of points around to rearrange the pieces relative to one another you have no notion of measurability on the pieces, they have no notion of 'volume' or 'density'. If you put them back together in just the right way, however, you can construct a new arrangement which does have a well defined notion of measurability, of volume or density. The theorem shows that it is possible to obtain a new configuration which is a double copy of your original configuration and when you compute the sum of their measures you get twice the amount.

Though it is a terrible bastardisation to put it like this you could perhaps view it as starting with a ball of uniform density of volume 1, cutting it in such a way that each 'piece' has zero volume (being a horrific configuration of points not joined up into larger structures like slices), these are then rearranged to give double the volume, zero, and then rearranged to reconstruct two copies of the ball, each with volume 1. Volume = 1 goes to Volume = 0 by cutting, Volume = 0 doubles to volume = 0 by rearrangement and then a pair of volume = 1 objects are reconstructed. I'm probably making set theorists break out into a cold sweat but that's one way to look at it (if you ignore any and all formalisation and many many relevant factors).

Now this might sound bizarre, given what we experience in the real world (as chinglu illustrates, some people cannot let go of their biases brought about by ignorance), but we're not talking about atoms or the like. The axiom of choice lets you split up the continuum of points which make up the initial sphere in a way what does not adhere to the same rules as counting finitely many indivisible pieces, as would happen if you tried this with an apple. When you start trying to put infinitely many objects in infinitely many sets, particularly if you start with an uncountable set lots of complicated and counter intuitive things can occur. The applicability (or lack thereof) of many of these concepts to physics does nothing to diminish the validity of the result from a mathematical point of view. Though the B-T paradox doesn't arise in mathematical physics set theory forms the formal basis of a great many mathematical tools within physics. And before chinglu tries to use that statement to declare something is flawed within mathematical physics as a result this isn't an all or nothing thing, some part of set theory being conceptual, not physical, doesn't mean ALL of set theory is without physical applicability.

 

[someguy1]

Also, since non-measurable sets are sets, does this imply ZFC sets are not compatible with physical reality?

Even ZF (excluding the Axiom of Choice) is not compatible with physical reality. The Axiom of Infinity states that there exists an infinite set. That's not compatible with physical reality as far as anyone knows.

Not only that, the infinite set guaranteed by ZF to exist is a model of the Peano axioms for the natural numbers. Nobody, not even proponents of "infinitely many universe" speculation, believes that the natural numbers have physical existence.

http://en.wikipedia.org/wiki/Axiom_of_infinity