Bill Thurston's answer to “What's a mathematician to do?” (2010)

Re:Bill Thurston's answer to “What's a mathematician to do?”... by User23 on Monday June 08, 2020 @11:09PM (#23463246)

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?

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Re:Bill Thurston's answer to “What's a mathematician to do?”... by verma7 on Tuesday June 09, 2020 @12:10AM (#23463558)

|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.

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Re:Bill Thurston's answer to “What's a mathematician to do?”... by roenxi on Tuesday June 09, 2020 @04:28AM (#23464834)

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|.

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