Cantors diagonalization argument

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I am sure you are all familiar with this. The number generated by picking different integers along the diagonal is different from all other numbers previously on the list. But you could just put this number as next element on the list. Of course that just creates a new number which is missed, but if you successively kept putting the number missed as indicated by the diagonal wouldn't you eventually hit all real numbers on the interval (0,1)? I mean isn't it the same as saying that we haven't hit the rational number 5/32 but that it is coming later in the sequence we use to pair the rationals with natural numbers.

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"I am sure you are all familiar with this. The number generated by picking different integers along the diagonal is different from all other numbers previously on the list. "

Partially true. Remember, you made the list by assuming the numbers between 0 and 1 form a countable set, so can be placed in order from smallest to largest, and so your list already contains all of those numbers.
Now, when you go down the diagonal to create the new number, the procedure discussed does not stop at any particular row (say the 500th). Instead, it shows you can go down all rows, creating a number that is different from EVERY other number in your (assumed to be complete) list. So now you have this:

* You assumed you could list every possible number, and that you have done so
* You find out your assumption was wrong

That is the key to the argument.