TITLE: Architecture of Cantor's proof
QUESTION [5 upvotes]: Cantor's diagonal argument consists of two parts: bijection and the extraction of the new number. If he shows that a given architecture of bijection doesn't work, why does it imply that any other architecture of bijection should not work either?
Just an addendum to make my point clearer:
Another possible architecture that comes to my mind is to write real numbers in the table and correspond them to the nth power of 2. Then we start to construct new numbers that weren't listed in our table and correspond them with powers of 3. After that we construct new numbers from the table of power of 3 (even if without checking we had them in the first table) we correspond them to the nth power of 5, and so on and so forth.
It is easy to notice that this is a poor bijection architecture as you can put them back into one list and just repeat the construction of the new number.
Why can it be mapped back to a simple list for any architecture?
On the other hand we can easily think of correspondence of natural numbers to itself, in a way that we will end up having extra unlisted numbers.E.g.(1->2,2->4, etc...). Obviously it doesn't imply that there are more natural numbers than 'natural numbers'.

REPLY [7 votes]: Long comment
Can be useful to read Cantor's original proof of the theorem :

There are infinite sets that cannot be put into one-to-one correspondence with the set of positive integers

into : 

Georg Cantor (1892), "Ueber eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht der Deutschen Mathematiker-Vereinigung 1890–1891, 1: 75–78 :


Let consider a set $M$ of elements of the form $E = (x_1, x_2, \ldots, x_{\nu}, \ldots)$ where each "coordinate" $x_i$ is either $m$ or $w$.

If $E_1, E_2, \ldots, E_{\mu}, \ldots$ is any infinite list [unendliche 
    Rehie] of elements of the set $M$, then there is always and element $E_0$ of $M$ that does not match any $E_{\nu}$ [keine $E_{\nu}$ übereinstimmt].


The "diagonal argument" follows.
The key-points of the proof are :

its generality : introducing sequences of abstract symbols, Cantor shows that the uncountability is not depending on some specific property of real numbers
the proof is "constructive" : for any given list, it gives us a "procedure" to manufacture a new element not in the list.


See also :

Robert Gray, "Georg Cantor and Transcendental Numbers", American Mathematical Monthly, 101: 819–832.