Unbounded real over a model
= Unbounded real over a model
{title2=$f\not\leq^*g\quad(g\in M\cap\omega^\omega)$}
A function $f\in\omega^\omega$ is unbounded over a model $M$ if for every $g\in M\cap\omega^\omega$, infinitely many $k$ satisfy $g(k)<f(k)$. Equivalently, no ground-model function eventually dominates it. This is weaker than being a <dominating real> over $M$.