4 ms·

That's beautiful -- thank you for sharing the post. I don't get to think about pure math very often any more, but this was a wonderful reminder of clarity in a

by ISL 6y ago

That's beautiful -- thank you for sharing the post. I don't get to think about pure math very often any more, but this was a wonderful reminder of clarity in a world that's a little more disordered than usual. One of the little joys of math, well known, but always makes me smile: Though there are infinitely many rational numbers, they are countable.

JadeNB 6y ago
> One of the little joys of math, well known, but always makes me smile: Though there are infinitely many rational numbers, they are countable. I think 'infinite but countable' isn't so surprising—even without a formal definition, I think most people would expect the counting numbers to be countable—but maybe the fact that there are infinitely many more rational numbers than counting numbers, and yet there are exactly as many rational numbers as counting numbers? (This is one of many ways of phrasing it, but it perhaps understates how much bigger the rationals seem to be than the counting numbers; the description I've given would apply as well to the set of all integers, whose countability is still perhaps surprising, but not as surprising.)
- User23 6y ago
  There's one for each of 'em, to use an idiom. A bijection exists between the Z and Q. Both sets are exactly the same size. Cantor's argument is certainly quite clever, but "there are infinitely many more rational numbers than counting numbers, and yet there are exactly as many rational numbers as counting numbers" is nonsensical to me. Am I missing something?
  - roenxi 6y ago
    Posting in agreement with you, I think it is more intuitively satisfying to say something like "|X| is the smallest well-defined mathematical object that is >= the size of X. |Q| = |Z|". It isn't intuitively satisfying to say that there are "as many" rationals as integers because that is obviously not true; there are multiple rationals between any two integers. An argument to ignore that pattern as we scale up to infinity is rather flimsy. But if we talk about infinite sets we are forced to admit the existence of well defined things that are plainly larger than |Z|, and no such well defined objects that exist between |Z| and |Q|.
  - verma7 6y ago
    |Q-Z| is countably infinite, but |Q| = |Z|. There are countably infinite rational numbers that are not counting numbers, yet there are exactly as many rational numbers as counting numbers.

by ISL 6y ago

JadeNB 6y ago

> One of the little joys of math, well known, but always makes me smile: Though there are infinitely many rational numbers, they are countable. I think 'infinite but countable' isn't so surprising—even without a formal definition, I think most people would expect the counting numbers to be countable—but maybe the fact that there are infinitely many more rational numbers than counting numbers, and yet there are exactly as many rational numbers as counting numbers? (This is one of many ways of phrasing it, but it perhaps understates how much bigger the rationals seem to be than the counting numbers; the description I've given would apply as well to the set of all integers, whose countability is still perhaps surprising, but not as surprising.)

User23 6y ago
There's one for each of 'em, to use an idiom. A bijection exists between the Z and Q. Both sets are exactly the same size. Cantor's argument is certainly quite clever, but "there are infinitely many more rational numbers than counting numbers, and yet there are exactly as many rational numbers as counting numbers" is nonsensical to me. Am I missing something?
- roenxi 6y ago
  Posting in agreement with you, I think it is more intuitively satisfying to say something like "|X| is the smallest well-defined mathematical object that is >= the size of X. |Q| = |Z|". It isn't intuitively satisfying to say that there are "as many" rationals as integers because that is obviously not true; there are multiple rationals between any two integers. An argument to ignore that pattern as we scale up to infinity is rather flimsy. But if we talk about infinite sets we are forced to admit the existence of well defined things that are plainly larger than |Z|, and no such well defined objects that exist between |Z| and |Q|.
- verma7 6y ago
  |Q-Z| is countably infinite, but |Q| = |Z|. There are countably infinite rational numbers that are not counting numbers, yet there are exactly as many rational numbers as counting numbers.