This is interesting. I understand the Axiom of Choice to be problematic in our mathematics and is impacting physics.

If your not above being spooked or letting me know what you think about my argument, then I'll say why exactly I think this.

The biggest problem about the Axiom of Choice is exactly what you mentioned was a good thing, 'ending up with more parts than you have elements'. Without a unique index value for each part then you can't identify them. This causes non-uniqueness problems in physical equations. Those elements are indexed but if you add more parts then you can't know the true dimensionality of your data/observations. The extra parts get nested underneath each other by our positional notation.

In physics we use dimensional analysis which is based on rational algebra. I am not sure if physicists realized yet that we use Real numbers and Rational numbers mixed together. The problem here is mixing two incompatible number systems. The Reals are an infinite set of infinite sets and is uncountable for that reason. Rational's are countable.

*When we map functions from real to rational we lose imbedded information data since that set is unquantifiable smaller* Not sure what happens in the other direction.

On the flip side though, this error gave us fractals and chaos so its not all bad.

Lastly, Real numbers simply aren't geometric. Most are irrational and infinite geometric series expansions. I am not sure how physics was able to resolve that since everything is supposed to be in the real world.