5 ms·

So what does it mean for physics? Perhaps zero, one, more or all of the following: - duplication of volume seems to duplicate mass and energy, which is imposs

• untilted 2y ago
  A physical sphere has a finite number of constituents, therefore the theorem does not apply
  • deleted 2y ago
    [deleted]
• mjburgess 2y ago
  None of these. The 'axiom of choice' isnt a property of the real world one way or another. The issue is that physical measures (length, area, volume, etc.) require measure theory in mathematics to specify properly. The issue is the mathematics was too simple to model the relevant physical systems. All of physics is specified in terms of a spacetime continuum, neither this, nor anything else, indicates that physics requires a revision.
• andrewla 2y ago
  The disconnect between reality and Banach-Tarski is deep. Important to the paradox is that it does not present a construction of the division. And more importantly, it relies on the fact that it is not possible to construct such a division. If you accept some notion of the Universal Church-Turing Thesis as a matter of reality, then it is not possible even in principle for such a division to occur. Note that this is not just a matter of "finding the right way to construct it" -- the theorem critically relies on the fact that given any way of constructing sets using finite representations (like computer programs or layout of molecules), the theorem does not hold.