3 ms·

> What axiom lets either of us do an infinite number of steps and say we're finished? The same one that says we can build the naturals from successors — withou

> What axiom lets either of us do an infinite number of steps and say we're finished? The same one that says we can build the naturals from successors — without that, we don’t have all of the naturals. As I’ve been saying repeatedly, this is due to building sets from induction. A limit L for an infinite sequence (an) is such that for any epsilon, there’s some n where after that n, | L - an | < epsilon. You have a set of rationals indexed by naturals building towards that real number — but you can do it the other way and define reals as those sets which converge to it. > And as you keep adding more digits, I can keep telling you an infinite number of places that match. The reals that are indescribable are truly weird: they look like subsets of N that are infinitely large but exclude infinitely many as well, without any pattern to inclusion or exclusion. And you can show this power set definition matches [0,1] by taking the binary 0 or 1 at position n to be if n is included in that subset. To get rid of the weird reals, you need to force sequences like that out. What makes the reals larger is that they’re defined to include the limits of all converging sequences. Rationals don’t, eg, sqrt(2) or pi. > But if we take turns, where I make a finite number of steps and you give me your next digit, then it's my turn again, I can keep matching you all day and all night. But this doesn’t end in a bijection, because after you follow this construction for an infinite number of turns, I can name a real number not in your list — as my final move. That’s what the schema says: no matter how we construct the list, after it’s built, I can name a number outside of it. You’re not showing the list is complete, ie includes all reals, merely that we can include a particular number if I tell you it digit by digit — but we knew that by construction (ie, there will always be some countable converging sequence of rationals). Every real is constructed by some countable sequence of digits; but there’s more real numbers than countable infinity.

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.

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.