Those are not notions from "set theory" itself but from "numerical dynamics" where we have some kind of recursion formula, say , or differential equation dy/dx= f(x,y). (The term "dynamics" is from the similarity to motion where the future position of a body is determined by its position now.)
If we are given some value, or we could use those formulas to determine or y(x) for different values of n and x. In particular, the "positive set" is the set of all or for k> 0 and x> 0. The "ultimate set" is the set of values for or in the limit as n and x go to infinity.