If you use this argument, then we can visit the uncountable argument, which seems to use uniqueness to identify numbers that are left out of the infinite row of numbers.
Or, for example, why couldn't I just do this with the sets E = {1,2,3...} and F = {1,2,3...}:
E|__|F
1-not-1
2-not-1
3-not-1
.
.
.
I don't think you're quite ready for the uncountable argument; first you need to really grok how "countable" infinite sets behave. Let's keep going on the integers.
(But the quick answer is that with finite sets which mapping you use never makes a difference as to whether you succeed or not, but with infinite sets sometimes you succeed and sometimes you fail. What you're doing with that example is exhibiting one line-up that fails -- it isn't a proper 121C because 1 never shows up in the F column. If these were finite sets, you'd be done -- when you show one failure you've shown that all the line-ups fail. I think that's where your intuition is letting you down: you think you're done because you're used to finite sets. Since these are infinite sets, you're not done. You've shown one example of a line-up that has a missing set member on one side; but you haven't shown that
every line-up has a missing set member. (And of course we know there's a line-up with no missing members, since E and F are the same set.) But in Cantor's "diagonal" argument for the reals being an uncountable set, he shows that
every line-up has a missing real number.)
But doesn't the argument only show that the line-up of the naturals is missing real numbers?
Yes. But in your version,
you chose which line-up to use. You're the one who picked a line-up in which 1 doesn't match 1 and 2 doesn't match 1 and 3 doesn't match 1 and so forth. Somebody else could always come along and pick a different line-up, such this: (1->2, 2->1, 3->4, 4->3, ...). So 1 isn't missing from F in
her line-up. Nothing is missing; she's got a complete 121C between E and F.
In contrast, in Cantor's version of the argument, Cantor doesn't choose which line-up to use. Think of it like a chess game: it's easier to win if you go first. Choosing which line-up to use is the first move; showing that there's an element missing is checkmate. What you did was win by playing white and going first. What Cantor did was let the other guy go first, and then Cantor goes on to win even though he's playing black. His argument is "You pick the line-up. Here's how I show that whatever line-up you picked is missing a real number."
Who gets to go first makes all the difference -- it means that you've shown that
some line-ups from the first set to the second set are missing one of the second set's members, but Cantor's shown that
all line-ups from the first set to the second set are missing one of the second set's members.
But I thought that uniqueness matters. Each hotel room is the same, and each person is the same. In other words, I have no problem with an infinite number of eggs each sitting in an infinite number of nests.
No, the hotel rooms and the people are all different. At the very least, each hotel room has a different number on its door. The people are especially all different. Imagine if Hilbert has his infinite hotel with every room full, and the new customer shows up at the front desk, and Hilbert says "No problem, sir. I'll just put you in room 1, and I'll move the lady in room 1 into room 2, and I'll toss the gentleman in room 2 out into the snow, and I'll explain to him that it's okay because the lady who's now in his room is just the same as he is." Mr. Hilbert is going to have a very angry guest.
But this uniqueness seems to pop up in some ways but go away in other ways. For example, Cantor's diagonal argument. There is an issue with the "kind" of number that doesn't get listed, and aleph 0 won't eat it up. Then there is the kind of number that I don't want in aleph 0, but aleph 0 eats it up anyways.
It's not really a matter of what "kind" of number you're dealing with, but rather of how many of them there are. It's easy to have aleph 0 eat up exactly the kind of real number that Cantor showed was missing. Suppose somebody tries to win against Cantor by giving him the infinite list of the computable real numbers. Cantor uses his diagonal method and shows that there's a real number not on the list. Well, that means that the missing real number wasn't computable: there is no possible computer program for generating its digits one after another. (There are numbers known to be like that, such as "omega", also called "Chaitin's Constant".) But that doesn't mean somebody else couldn't make a list of aleph 0 uncomputable numbers. Here's aleph 0 eating up the same kind of number.
E G
1 omega
2 (omega + 1)
3 (omega + 2)
...
They're all uncomputable, and there are aleph 0 of them in the list. But of course (omega + pi) is also an uncomputable number and it's not in the list. There aren't any particular real numbers you can't put into a 121C with the integers; it's just that you can't use the same 121C for all of them. There are too many uncomputable numbers for any single list.
covering the entire sets:
