Let $f : X \to Y$ be a problem, for instance, $f: \Bbb{Z} \to $ a factor.

Given input measure $n = |x|$, then our problem is $O(g)$ if there exists an algorithm running on a standard machine, and an operation counting function $T(n)$, that counts the maximum number of operations on inputs $x \in X$ such that $|x| = n$ such that $T(n)$ is $O(g(n))$.

But how can you reckon this definition with integer multiplication, for example. There isn't a finite memory computer that can hold *all integers* so how can there be an algorithm that scales this way? Can you describe it as some sort of scaling occuring along an infinite sequence of growing-memory machines?