Is there a formal definition for iteration?

Question

I wanted know if there is a formal definition of iteration. In the sense that, this definition will say that after the 4th iteration is termination or the 5th iteration and not the 2nd iteration. Basically a rigorous definition of iteration.

Answer 1

The $k$th iteration is just the $k$th element of some appropriate sequence. In general, I doubt it's a formally defined concept; it's just one of those things whose meaning is basically the natural-language meaning of the word.

Answer 2

In the field of Denotational semantics they have developed a (very) formal definition of iteration in terms of recursive functions that you might like to look into.

https://en.wikipedia.org/wiki/Denotational_semantics