why is this a critique of measure theory? Measure theory is the answer to the paradox. The partition uses unmeasurable sets, so comparing the surface areas before and after doesn't make sense. You could do the partition a billion times and expand the volume as well..
> [Banach-Tarski] means there can be no measure satisfying the requirements even when weakened from countable additivity to finite additivity.
Measure theory was an attempt to rescue us from the breakdown of the Riemann integral for poorly-behaved functions by adding a ton of abstraction of formalism. And it turns out that it isn't even successful at that.
But it turns out that what "poorly behaved functions" means is basically "non-computable functions", and so you can back up even further and point to that idea as the problematic one in the classical foundations of set theory -- that constructibility is not only a nice-to-have but is in fact necessary for any sort of coherent theory.
Put differently, Banach-Tarski shows that formalism-based systems that rely on the axiom of choice and the excluded middle too heavily result in behaviors which are clearly incorrect, but which are internally consistent.
I'd like to point out this is at odds with most of the mathematical community's interpretation. Caveat: mathematics itself does not have concerns about "interpretation" and your interpretation is completely valid of course.
My interpretation is that powerful axioms often prevent theories which asks for "too much" because the power of the axioms then can form contradictions. For example asking for everything to be a set => Russel's paradox and asking for all sets to be measurable => Banach tarski type contradiction.
Side note for mathematically trained peeps: Of course assuming all sets to be measurable and the measure to be translation and rotationally invariant leads to much easier contradictions with AOC. Banach-Tarski gives a finite decomposition contradiction (vs say countable). Also not a analyst myself, so take everything here with a grain of salt. But honestly analysis is basically completely not possible without excluded middle so I think analysts would have opinions more on classical side than myself.
Right, but those were self imposed requirements based on intuition. The axioms of sigma algebras are different than these requirements. If anything measure theory highlights the limits and conditions of when to consider something measureable. That is the point of the Paradox and the need for measure theory.
> behaviors which are clearly incorrect, but which are internally consistent.
For mathematical objects, what other considerations are there than consistency?
I mean, physics has given us all manner of counterintuitive things. At what point do you stop saying "so much the worse for the theory" and start saying "so much the worse for my intuitions?"
There are certainly considerations other than consistency in mathematical theories. But I don't think I'll be able to articulate them well here. Maybe someone else can help.
