@everyone: focusing too much on some dynamic vs. static distinction is a red herring. It’s perfectly well plausible to understand the diagonalization argument even while “visualizing the process as a dynamic and growing thing” (see below).
@adhay: you said it’s your ignorance of mathematics which causes you to ignore some posts. But I think the best way to clear up your ignorance is to read some of those posts you’ve ignored, and I don’t think there are any posts in this thread which are beyond a novice layman’s ability to read. Could you please read and respond to this post, at least?
Let us put the argument like this: Bob comes up to you and says “Let’s play a game”. The idea behind the game is that Bob will win if he can craft a certain surjection, but you can win by showing that Bob hasn’t actually made a surjection.
The way it works is that there are four turns:
On turn 1, Bob gets to make up a square of colors. That is, Bob writes down some rules for how to answer questions of the form ‘What color (black or white) should go in Bob’s square at column #X and row #Y?’. Whenever someone asks a question like that, Bob’s rules should tell us a specific color to draw at the corresponding column and row of his square. In this way, if we liked, we could fill in the square bit by bit with colors, though, of course, it may be impossible to ever actually finish filling all of it in.
On turn 2, you get to make up a sequence of colors. That is, you write down some rules for how to answer questions of the form ‘What color should go in adhay’s line at column #X?’. Whenever someone asks a question like that, your rules should tell us a specific color to draw at the corresponding point in your sequence. In this way, if we liked, we could write out your sequence bit by bit, though, of course, it may be impossible to ever actually finish writing out the whole thing.
On turn 3, Bob tries to show that your sequence is actually one of the rows of his square. Bob does this by picking some row number and announcing “Bob’s row is #such-and-such”.
On turn 4, you try to show that Bob has failed. You do this by picking some column number and announcing “adhay’s’s column is #such-and-such”.
After turn 4, all that’s left to do is figure out the winner: we’ll follow Bob’s rules to figure out which color goes in Bob’s square at Bob’s row and adhay’s column. We’ll also follow adhay’s rules to figure out which color goes in adhay’s sequence at adhay’s column. If these two colors come out the same, then Bob wins. If they come out different, then adhay wins.
Would you be willing to play such a game, even if there was a thousand dollars riding on it?
Of course you would; that’s easy money! You can guarantee a win: on turn #2, just use the rules “The color in adhay’s sequence at column #X is the ‘opposite’ of the color in Bob’s square at column #X and row #X”. And on turn #4, pick adhay’s column number to be the same as Bob’s row number. The color that comes from adhay’s sequence here will be the opposite of the one that comes from Bob’s square here, and so you’ll win the thousand bucks.
Cantor’s theorem is nothing more than the assertion that you can guarantee a win at this game, and the diagonalization proof is nothing more than the above strategy for doing so.
Does that all make sense? Do you have any questions about or qualms with it?
[I know I haven’t mentioned real numbers anywhere above. That’s ok; like I said, you should forget about real numbers for now, they’re just a distraction and aren’t what the diagonalization proof is directly about]