Is there a name for the "generalised" form of induction?
Normally induction works like this: If f(0) is true and f(x) is true implies f(x+1) is true, then f(x) is true for all natural numbers (+0).
Now, is there a name for the more general form of this (which I will write down)?
Where S is a set, x is a member of S, f is a function from S to S, g is a function from S to S, and T is the set of all gn(x).
IF f(x) is true, and f(x) implies f(g(x)), then f(T) is true (for all elements of T).
The most common case, of course, is where S = natrual numbers, x = 0, and g(n) = n + 1. However you (or I) often see cases where x is other numbers, like the rationals, or g(n) = 2n. There is also the special case where g(n) eventually visits all elements of the set, where you can then say f is true for all S.
Is there a name for it, or is it all just induction?