4 ms·

"Countably infinite" makes zero sense to me. Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll ne

• feoren 3y ago
  > Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers. Exactly correct! This holds true of everything you can generate stepwise, even infinite sets. Cantor proved that you cannot "generate" (stepwise) all Reals between 0 and 1. Any infinite set you can generate stepwise is Countably Infinite. > I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem. Exactly correct! And then infinitely later, when you're "done", having generated every Real between 0 and 1, he will then generate a new Real not on your list. Oops! You have not generated all Reals between 0 and 1, even with infinite time. > And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going. Exactly correct! Any two sets you can pair together (via a bijection) have the exact same cardinality. Neither is more infinite nor countable than the other. Cantor proved you cannot "pair" the Reals with the Natural Numbers. You and Cantor agree completely. You're very close to understanding why the Reals are bigger.
  • dbtc 3y ago
    > And then infinitely later There can be no 'and then' after infinitely later. I don't see why stepwise is important but that must be the key to Cantor's proof. If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned. The ordering could be entirely random, I don't see how it makes a difference. There will always be enough integers to match. Is it that my black box metaphor is cheating by coercing a truly 'parallel' generation of decimals into a linear operation? But even then, if I'm getting exponentially bigger chunks of new decimals, I can provide equally large chunks of integers... so it still doesn't make sense to me. Infinity is infinity and you cannot count it.
    • ironSkillet 3y ago
      Mathematicians consider two sets to be of the same size or more precisely "cardinality", if it is possible to construct a 1-1 map of elements from the first set to the second set. These maps can obviously be constructed for sets with finitely many elements, and they can be constructed for sets with an infinite number of elements as well. For instance, the set of all integers has the same cardinality of the set of all positive integers (just enumerate the integers alternating back and forth expanding from 0 - this constructs the 1-1 map). We can prove that no such 1-1 map exists between the integers (an infinite set) and the decimals in the interval [0,1] (another infinite set). The proof is by contradiction, meaning that we assume such a 1-1 map exists and prove it leads to a contradiction, therefore our assumption that the 1-1 map exists must be false. So suppose we were able to construct a map from all decimals in [0,1], by enumerating them according to some clever rule. Let d_i be the ith digit of number i in your mapping. For each I pick another different digit d_i'. Let's construct the number with decimal representation D = . d_0' d_1' d_2' ... Assuming we have our 1-1 map, it must be somewhere in our mapping. Let's say it's element k. By our labeling concention the kth decimal digit of D is actually d_k. However, this contradicts our method of construction of D. Therefore our assumption that there is a 1-1 map between decimals in [0,1] and the integers must be false. It is in this sense that there are infinities of different sizes.
      • catiopatio 3y ago
        > It is in this sense that there are infinities of different sizes. They aren’t actually different sizes, though. All this proves is that under specific set theoretic assumptions, a contradiction arises if you define “size” as “cardinality” and assume that a particular bijective relation exists between your two infinite sets. It doesn’t actually mean the sets have different sizes, it just means they differ under a set of assumptions that may (or may not) be useful for your purposes.
        • deleted 3y ago
          [deleted]
        • ironSkillet 3y ago
• amanwithnoplan 3y ago
  I think "countably infinite" makes no sense to you because you have a different idea of countable than a mathematician (disclaimer: not a mathematician). A mathematician compares the size of two sets of stuff by pairing off items from each set, but this is not a mechanical process taking a finite or even unbounded amount of time: they just need to show such a mapping exists or that nonexistence would lead to a contradiction; they don't need to actually carry out the process mechanically. By definition (according to mathematicians), something is countable if it is the same size as the set of natural numbers {0, 1, 2, ...} or smaller, and "countably infinite" just means it is the same size as the naturals (and not smaller, which would make it finite). A small minority of mathematicians hold the position that proof-by-contradiction is not good enough, and that you really do need to positively prove something. They are called intuitionists. Presumably, an even smaller minority of mathematicians hold the position that this proof must (theoretically) be able to be carried out in a mechanical manner. They're some flavor of constructivists, but maybe they're better called programmers. <- This is where you are.