One thing which the author of this essay gets right is that contemporary mathematics is a bit dogmatic when it comes to the logical foundations. We should be looking at the Banach-Tarski theorem as evidence that a particular logical framework for reasoning about volumes/probability is more complicated than strictly necessary. Presumably no non-measurable sets exist in the "real world". At this point there are two possibilities:
- A theory with non-measurable sets is much simpler/more expressive than the alternatives and thus is still a useful tool.
- There is a simpler theory without this defect, which is at least as expressive when reasoning about real world phenomena.
In this case, the latter possibility turns out to be true. It is just unbelievably difficult to convince mathematicians to change the rules of the game, even if you can point to concrete gains.
Without the axiom of choice you get similarly bizarre results. For instance wothout choice there exists a surjection from the reals to a set of greater cardinality.
The existence of the reals is independent of the axiom of choice. The collection of mathematicians that don't believe infinite sets exist has maybe a few members. Constructive mathematicians believe the reals exists. They can be constructed.
...(maybe skip to chapter 5 to get to the meat of it). So constructivists don't believe in those types of numbers, and so get "choice" as a theorem instead of an axiom.
Constructive mathematicians definitely do believe the reals exist and are uncountable. Not all reals are computable or definable. The set of computable reals is countable. There are different varieties of constructive mathematics. A very large majority of constructive mathematicians believe the reals are uncountable. In intuitionistic mathematics the reals are uncountable.
In some constructive versions of math you get weird things like there being an injection from R to N but there not being a bijection due to diagonalization.
I think you are confusing computable/nameable with constructive as that term is used by most mathematicians.
What do you mean? If ZF without choice shows that a statement P holds then certainly ZFC also proves the same statement. Do you mean that there is a consistent extension of ZF with the statement "There exists a surjection from R to P(R)"?
There is a theorem that states that either there is a nonmeasurable subset of the reals or that there is a surjection of the reals to a larger set. The theorem is by Sierpinski. This is way out of my area but I'm guessing ZF is not strong enough to show either all sets are measurable or the existence of a non measurable set.
The common criticism of AC is Banach-Tarski. So if you don't agree with Banach-Tarski and want every subset of R to be measurable then you have to conclude an equally bizarre result.
As an algebraist I accepted AC. Instead of constantly saying, "let V be a vector space over k with a basis" it's easier to just assume all vector spaces have a basis. I think analysts need AC more than other branches, The Intermediate Value Theorem isn't provable without AC. I think.
What I found after Googling is that there is a theorem by Sierpinski which shows that if we make an additional assumption about the cardinality of [R]^\omega, which seems to be approximately the same as the boolean prime ideal axiom, then there is a non-measurable set. Thus the existence of non-measurable sets is implied by principles weaker than full choice, but that's not too surprising.
The case of measure theory is particularly interesting, though, since you don't even need to change your underlying logic to get a better model. For instance, if you base your "measure theory" on valuations on locales instead of measures on sigma algebras then the theory itself becomes simpler and the Banach-Tarski "paradox" goes away.
Briefly, in locale theory, your "measure" is defined on sublocales instead of subsets. While there are more sublocales than subsets, the condition for when two sublocales are disjoint is stronger. This is what breaks the Banach-Tarski construction. The orbit subsets used in Banach-Tarski still exist, but while they are disjoint as sets, they are not disjoint as sublocales and thus don't decompose the volume of the sphere.
I'm not familiar with locale theory. Thanks for the reference. I'm guessing there will be some non-intuitive results. My non-expert impression is that whatever one chooses in terms of logic and set theory there will be bizarre results when dealing with sets of cardinality of the continuum and not dealing with the continuum leaves out too much.
> There is a simpler theory without this defect, which is at least as expressive when reasoning about real world phenomena.
This is really interesting to know, and was exactly the type of think that I was wondering about. And I assume the author felt similarly although didn't draw identical conclusions.
Out of curiosity could you describe or link me to this other theory?
I was thinking about categorical probability theory or measure theory based on locales. Unfortunately there are (to the best of my knowledge) no textbooks or good writeups available for either, merely long lines of research papers. As said, it's a bit of a niche area, since most mathematicians don't want to think about reworking foundations.
- A theory with non-measurable sets is much simpler/more expressive than the alternatives and thus is still a useful tool.
- There is a simpler theory without this defect, which is at least as expressive when reasoning about real world phenomena.
In this case, the latter possibility turns out to be true. It is just unbelievably difficult to convince mathematicians to change the rules of the game, even if you can point to concrete gains.