Since the Banach-Tarski paradox makes a statement about domains defined in terms
ID: 1263873 • Letter: S
Question
Since the Banach-Tarski paradox makes a statement about domains defined in terms of real numbers, it would appear to invalidate statements about nature that we derived by applying real analysis. My reasoning is this:
If you can "duplicate" an abstract 3-dimensional ball defined, in the usual way, using the domain of real numbers, then clearly the domain of real numbers must be unsuited to describing physical objects (for instance The Earth), because duplicating them would double their mass (and thereby energy). But we use real numbers to reason about nature and derive new laws (or basic results) all the time in physics.
In the same way as we could integrate over the earth's volume to determine its mass from its density (for instance), we could apply the Banach-Tarski theorem to show that we could generate as many earths as we liked, right?
Explanation / Answer
Philosophers and engineers have often thought that the notion of a real number is, indeed, unphysical, but this has nothing to do with the Banach-Tarski paradox.
One can find seven pieces of a sphere which, when re-assembled, produce a sphere twice as large. That is not strictly speaking a paradox, but it certainly violates one's geometrical intuition and must be un-physical.
The key point is that these pieces are not measurable: their boundaries are so complicated that one cannot meaningfully talk about their volume. It follows from this that the pieces cannot even be approximately produced by physically implementable cutting or slicing or powdering operations.... The notion of measure, as in probability theory, integration theory, and real analysis, is rather a mathematical technicality, so I won't go into it unless requested, but if you think of volume, you will get the right idea.
I will say that in more mathematically rigorous treatments of chaotic dynamics, such as Benatti, Narnhoffer, Thirring, also Connes, the physical dynamical variables, or observables, must be measurable functions on the phase space.