> 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.
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.