4 ms·

I won't claim to have any proof in hand, but there's still something fishy about it all. I think things like Banach-Tarsky and such ought to be treated as coun

by xscott 2y ago

I won't claim to have any proof in hand, but there's still something fishy about it all. I think things like Banach-Tarsky and such ought to be treated as counter examples to show the absurdity of the Reals. At times I've thought maybe the problem with diagonalization is forcing me to do a (countably) infinite number of steps before Cantor gets to take an infinite number of steps. For instance, I can give a trivial counting scheme and claim it will eventually generate every number between zero and one, perhaps by writing a program to write programs that generate digits. It's all very deterministic, and I state exactly what I'm doing up front. So I give the first number (program), and Cantor gives his first digit and says it's not in my list. Then I tell Cantor the countably infinite number of places where his first digit is found, and we select those programs. Then he gives me his second digit, and I select the subset where those are found. So we go back and forth, he keeps saying his number so far is not in my list, and I keep telling him all the places where it is in the list... Feels like stalemate there. There are other places (such as summing the alternating harmonic series) where you have to be very careful about the ordering of operations. If he really has a higher order infinity of numbers not in my list, maybe I should be able to ask him for his list first, no? It's uncountably infinite, but he can't provide even one of them.

zmgsabst 2y ago
You don’t have to do anything but assume such a bijection does exist, as in my comment: its existence leads to a contradiction in the presence of limits — just like I showed. But without that same limiting to build sets, you can’t construct N from the successor function and you can’t discuss the power set of N. So we either need to throw out countable limiting to build sets or accept that such set building rules out bijections between reals and naturals. You think Banach-Taraki is weird (as do most people), but I’d argue a world where you have finite numbers or couldn’t discuss the set of subsets of the naturals is even weirder. > maybe I should be able to ask him for his list first, no? There’s no such list indexable by the naturals, as shown by that contradiction. If you allow indexing by the reals, that request is trivial.
- xscott 2y ago
  You've edited your post a bit from what I originally replied to. Looking at your bijection stuff, I think you're just restating Cantor's diagonalization with `(a_n + 1) % 10` as your mechanism for choosing digits, but maybe I've got that wrong. I appreciate that it's well defined, but I think it ignores my complaint. Let's say I tell you that I'm building my list of numbers between zero and one by taking the natural numbers, reversing the digits, and just putting a decimal place in front: 0: 0.0.... 1: 0.1.... 2: 0.2.... ... 10: 0.01... 11: 0.11... 12: 0.21... It's a silly mapping, and I'm sure there are a few problems with it, but let's start there. For now it doesn't matter what digits I'll choose afterwards - maybe it's all zeros, maybe it repeats the digits, maybe I'll do something more clever, but skip that for now. My list has a number that agrees with your number to any number of digits we choose. You and Cantor have to force me to give my infinite list of numbers all at once, and then build your infinite length counter example all at once. What axiom lets either of us do an infinite number of steps and say we're finished? What axiom says you can do an infinite number of steps after I do my infinite number of steps? When we do limits with deltas and epsilons, we say that we will get closer as we keep going, not that we got there. If you give me the first 1000 digits of your counterexample, I can tell you all the elements in my set that match your number up to the first 1000 digits. If you give me 1001, I can tell you all the places that match that too. A million, a gazillion, I can keep matching. And as you keep adding more digits, I can keep telling you an infinite number of places that match. So it really seems like there's a problem with the order of operations. It's like playing chess, and you force me to make all my moves and come afterward to checkmate me. 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.
  - zmgsabst 2y ago
    > 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.

by xscott 2y ago

zmgsabst 2y ago

You don’t have to do anything but assume such a bijection does exist, as in my comment: its existence leads to a contradiction in the presence of limits — just like I showed. But without that same limiting to build sets, you can’t construct N from the successor function and you can’t discuss the power set of N. So we either need to throw out countable limiting to build sets or accept that such set building rules out bijections between reals and naturals. You think Banach-Taraki is weird (as do most people), but I’d argue a world where you have finite numbers or couldn’t discuss the set of subsets of the naturals is even weirder. > maybe I should be able to ask him for his list first, no? There’s no such list indexable by the naturals, as shown by that contradiction. If you allow indexing by the reals, that request is trivial.

xscott 2y ago
You've edited your post a bit from what I originally replied to. Looking at your bijection stuff, I think you're just restating Cantor's diagonalization with `(a_n + 1) % 10` as your mechanism for choosing digits, but maybe I've got that wrong. I appreciate that it's well defined, but I think it ignores my complaint. Let's say I tell you that I'm building my list of numbers between zero and one by taking the natural numbers, reversing the digits, and just putting a decimal place in front: 0: 0.0.... 1: 0.1.... 2: 0.2.... ... 10: 0.01... 11: 0.11... 12: 0.21... It's a silly mapping, and I'm sure there are a few problems with it, but let's start there. For now it doesn't matter what digits I'll choose afterwards - maybe it's all zeros, maybe it repeats the digits, maybe I'll do something more clever, but skip that for now. My list has a number that agrees with your number to any number of digits we choose. You and Cantor have to force me to give my infinite list of numbers all at once, and then build your infinite length counter example all at once. What axiom lets either of us do an infinite number of steps and say we're finished? What axiom says you can do an infinite number of steps after I do my infinite number of steps? When we do limits with deltas and epsilons, we say that we will get closer as we keep going, not that we got there. If you give me the first 1000 digits of your counterexample, I can tell you all the elements in my set that match your number up to the first 1000 digits. If you give me 1001, I can tell you all the places that match that too. A million, a gazillion, I can keep matching. And as you keep adding more digits, I can keep telling you an infinite number of places that match. So it really seems like there's a problem with the order of operations. It's like playing chess, and you force me to make all my moves and come afterward to checkmate me. 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.
- zmgsabst 2y ago
  > 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.