[maze-le]

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.

[tsimionescu]

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.

[bubblyworld]

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!

[petters]

Really? I think this is on a completely different level of intuition. There are five pieces here that are only rotated and translated.

[stared]

"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".

[Xcelerate]

> 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.

[IngoBlechschmid]

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.

I included some references about this in a set of slides: https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...

[dist-epoch]

My understanding is the general consensus is that no physical infinity can exist.

So Axiom of Choice/Banach-Tarski doesn't really apply in physics since they are only interesting when talking about infinite sets.