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> The reals that are indescribable are truly weird [...] Well, I appreciate your patience. I'm certain I won't find a hole in any of your arguments or the con

by xscott 2y ago

> The reals that are indescribable are truly weird [...] Well, I appreciate your patience. I'm certain I won't find a hole in any of your arguments or the conventional wisdom, but the Reals outside of the Computables (or Defineables might be a better choice) are more than weird, they're absurd. They're dense in the number line but only a countable number of them are anywhere to be found, named, or described past a few digits. Additionally there is always a Computable number between any two Reals and a Real between any two Computables. *This* ought to imply a one-to-one correspondance, but I'm certain I can't defend that argument either. As for applied math, any situation where we claim the Reals are required (perhaps as the position of a particle or the coefficients of a wave function or something) is uncomputable. So saying the Reals apply to reality probably says something very weird about determinism too.

zmgsabst 2y ago
> 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.

by xscott 2y ago

zmgsabst 2y ago

> 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.