This must be the most unintuitive result of all of mathematics. Its very interesting what a seemingly simple axiom like the axiom of choice can lead to -- simple as in 'even a 9-year old can understand it', the consequences are rather enormous and not simple at all.
Honestly, it's not that surprising if you learned the properties of infinity before, especially of uncountable infinity. If 2*Inf == Inf, and if a sphere has an infinity of points, it's not that surprising that you can make two spheres from those same points. The construction itself is of course much more impressive, I'm not downplaying it, but I don't think it's less intuitive than other properties of infinity.
My personal reckoning with this was learning that there are as many numbers in the [0,1] interval of the real line as on the whole real line.
The BT paradox includes the requirement that the pieces are separated and put back together using isometries of R3, which is _way_ more restrictive than isomorphism of sets (what you're talking about). So it's quite surprising from that point of view!
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
Well, the axiom of choice gives a lot of counterintuitive examples, with the Banach-Tarski paradox being the easiest to imagine by a non-mathematician.
Yet, I know no consequences that would be measurable in physics. To my knowledge, AoC is more like glue, which (paradoxically) makes quite a few things smoother, e.g., all Hilbert spaces have a basis. Otherwise one runs in a lot of theorems, in all corners of maths, with "this is always true for finite, for infinite we know that there are no counterexamples, yet we cannot prove that for all cases".
> Yet, I know no consequences that would be measurable in physics
One can build a physical device modeled off of a Turing machine that enumerates all proofs within ZFC. The machine halts if an inconsistency is discovered, and runs forever if not. Now a prediction can be made about a process in the physical universe whose outcome depends on the axiom of choice.
I’m not trying to sound facetious actually. Highly abstract mathematics plays a critical role in inductive inference (in the sense of speeding up universal search by mapping a search over program space to a search over proofs in formal systems). This appears to be the direction some recent ML research is heading, so it wouldn’t surprise me if a lot of “unphysical” axioms end influencing our ability to efficiently approximate Solomonoff induction.
Perhaps ironically, despite appearances, the process you propose does not depend on the axiom of choice.
This is because we can prove, in the small and generally trusted metatheory PRA, that ZFC is inconsistent if and only if ZF (= ZFC − AC) is inconsistent (if and only if IZF (= ZFC − AC − LEM) is inconsistent).
[ This metaproof rests on the fact that ZF can prove that the axiom of choice (AC) holds in "Gödel's sandbox" L, the "constructible universe", even if it might not hold in the universe of all sets. ]
In other words: Adding the axiom of choice to ZF doesn't cause new inconsistencies. In case ZF is consistent (a statement which most logicians believe), then ZFC is so as well.
Indeed, and one can give specific metatheorems in this direction:
For instance, regarding statements of the form "for all natural numbers x, there is a natural number y such that %", where in "%" no further quantifiers appear, there is no difference between ZFC (Zermelo–Fraenkel set theory with the axiom of choice), ZF (set theory without the axiom of choice) and IZF (set theory without the axiom of choice and without the law of excluded middle).
Any ZFC-proof of such a statement can be mechanically transformed to an IZF-proof, with just a modest increase in proof length.
As far as we can tell, GR implies, and we have measured, space-time is completely continuous. Draw a square on a piece of paper; or, better yet, outline a cube with some sticks: within that square (or cube) is an infinite set of points of either the integral or real cardinality — whichever you’d like.
The “no physical infinity” thing sounds like a very Greek sort of axiom — like their “nature abhors a vacuum” thing, etc.
To my mind GR (or at least the standard textbook version of it, anyway) _assumes_ that space-time is continuous, it does not _imply_ it as such. Continuity is baked into the foundations of any physical theory that is expressed in the language of differential equations.
You probably know this, but it's easy to confuse the map (a physical theory) with the territory (reality, which is far more complicated).
No, physicists do not think that space is like a tinily subdivided grid. Lots of physics would break if that were the case, including QM.
We have a lot of hints that the amount of information in any given space might be bounded, but yet that space appears continuous. How exactly you reconcile this is (one of) the mystery of quantum gravity.
Sure, we cannot measure infinity, but to be fair, all mathematical concepts (when looked at closely enough) are not something we measure directly.
Even if a kindergarten-level maths of "there are three apples," we do an abstraction. We need to decide that something is a separate object, an apple (how big or small should a fruit be an apple? if there is a bite, is it an apple? etc, etc) - usually with an assumption that all apples are the same (which we know is not true, but serves as an useful approximation).
pretend that
> David Hilbert famously argued that infinity cannot exist in physical reality. The consequence of this statement — still under debate today — has far-reaching implications.
