> 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.
Exactly correct! Any bijection between the naturals and the reals would suffice to show that they're the same cardinality; the order does not matter. I think where you're getting confused is just in who's trying to do what; who's the "protagonist" and "antagonist" in the proof.
Cantor is not trying to overwhelm you with so many real numbers that you run out of integers. Instead, he completely accepts and agrees with everything you're saying. And then he says: okay, pick any numbering of the reals you like. 1.11 is 4, 1.111 is 76, and 1.1111 is 445662323. It doesn't matter. You pick the pairing. Write your pairing down on an infinitely long sheet of paper. If the reals and integers have the same cardinality, there must be some way to write them all down on an (infinitely long) list. Pick any one and write it down.
Cantor's only job now is to show you that any real number exists that is not on your list. To do this, he constructs a number a digit at a time. He looks at the 1st digit of the 1st number, and writes down a different digit for his 1st digit. He looks at the 2nd digit of the 2nd number, and writes a different one for his 2nd digit. He looks at the nth digit of the nth number and writes a different one down for that digit, for every digit. Real numbers never run out of digits, so this goes on forever.
If this number he has written down is on your list, you should be able to point to a number on your list and say "Aha! You see, that is just real # 65,334,649!" but you can't, because it's different from that number in its 65,334,649th digit. It is truly different from every number on your list. And so there are more reals than integers.
I feel the number he generates via any procedure would be on the list since all of them are on the list. What am I missing about the plausibility of a procedire that must be possible which must generate a real number not in the series?
> I feel the number he generates via any procedure would be on the list since all of them are on the list
The claim is that all of them are on the list. The constructed number proves that claim false.
It's a proof by contradiction. If you assume there is any way to write an infinite numbered list of all reals, then Cantor shows it's possible to come up with a number not on your list. The construction uses your list as input, and given any list, can always produce a real number not on that list. Therefore there is no way to write an infinite numbered list of all reals.
It relies on the fact that real numbers have (countably) infinite digits, and therefore infinite "degrees of freedom" to be different. This may be one reason it's hard to accept. A "true" real number can contain infinite information in a single number. For instance, we can jam all of the naturals into a single real by just concatenating their decimal representations: 0.1234567891011121314151617181920212223...
This one single real number encodes the full infinite natural number line. That hopefully gives you a sense of why the "infinite digits" definitions of reals makes them qualitatively "bigger" than any number that has finite representation.
No I still don't get it, it's like saying that infinity^2 is larger than infinity. If 0^2 is no larger than 0, then it must be the same for infinity.
I see how a list of reals is like 2D list of infinities, so one grows from the middle and the other grows from the end, but they're both still infinite. I guess I'm still stuck in a 'mechanical' approach and not a mathematical one. I'm not sure I want to leave ;) This has been fascinating to think about anyway.
Exactly correct! Any bijection between the naturals and the reals would suffice to show that they're the same cardinality; the order does not matter. I think where you're getting confused is just in who's trying to do what; who's the "protagonist" and "antagonist" in the proof.
Cantor is not trying to overwhelm you with so many real numbers that you run out of integers. Instead, he completely accepts and agrees with everything you're saying. And then he says: okay, pick any numbering of the reals you like. 1.11 is 4, 1.111 is 76, and 1.1111 is 445662323. It doesn't matter. You pick the pairing. Write your pairing down on an infinitely long sheet of paper. If the reals and integers have the same cardinality, there must be some way to write them all down on an (infinitely long) list. Pick any one and write it down.
Cantor's only job now is to show you that any real number exists that is not on your list. To do this, he constructs a number a digit at a time. He looks at the 1st digit of the 1st number, and writes down a different digit for his 1st digit. He looks at the 2nd digit of the 2nd number, and writes a different one for his 2nd digit. He looks at the nth digit of the nth number and writes a different one down for that digit, for every digit. Real numbers never run out of digits, so this goes on forever.
If this number he has written down is on your list, you should be able to point to a number on your list and say "Aha! You see, that is just real # 65,334,649!" but you can't, because it's different from that number in its 65,334,649th digit. It is truly different from every number on your list. And so there are more reals than integers.