Questions about Cantor's diagonalization argument

[Radar]

Ah, i may have confused induction with iteration. What do you mean then by iteration?
And I probably can't give you the technicalities. I used this only once in a lecture i heared and that was like ten years ago.

Iteration as in any process you go over step by step constructing something. In this case it was about extending some initial set one element at a time.

Unfortunately I'm not aware of any good accessible explanations of transfinite recursion, short of just reading a set theory textbook. John Carlos Baez's series on large countable ordinals could be helpful if you aren't familiar with ordinals numbers, though as the title implies it stops short of what we need, and doesn't directly address recursion.

I think that knowing the specific term of transfinite recursion will be enough for finding all definitions. Thanks. :)