> are more than weird, they're absurd They’re a consequence of subsets being more complicated to talk about than the set itself. Real numbers are weird compar

> are more than weird, they're absurd They’re a consequence of subsets being more complicated to talk about than the set itself. Real numbers are weird compared to naturals because the indescribable ones match to subsets with infinite complexity. But functions on the real numbers are similarly large compared to the reals themselves — because they can have any subset of reals as their output, including infinitely complex ones. > This ought to imply a one-to-one correspondance, but I'm certain I can't defend that argument either. The interval is like a fractal: between any two points is a full copy of it. And for any prefix there’s a lot more “and then the tail is infinitely weird” than coherent tails. > So saying the Reals apply to reality probably says something very weird about determinism too. Math is a model. Nothing we compute can’t be don’t purely over computable numbers — but you simplify a lot of proofs if you work in the space where every convergent sequence has a limit, then just approximate. Including proving that approximations work nicely. You’re not making a claim about reality; you’re simplifying your mathematical machinery by packing all of the badness into a few technical axioms about induction — and then just shrugging because Banach-Tarski paradoxes are probably unphysical and purely a construction of the model.