This definition is part of Hovey's book on model categories:
Let $\gamma$ be a cardinal. An ordinal $\alpha$ is $\gamma$-filtered if it is a limit ordinal and, if $A\subseteq \alpha$ and $|A|\leq \gamma$, then $\sup A<\alpha$.
Why isn't this just saying that an ordinal $\alpha$ is $\gamma$-filtered if it is a limit ordinal and $|\alpha|>\gamma$?
If you take $A\subseteq \alpha$ and $|A|\leq \gamma$, then $\sup A$ is an ordinal with $\sup A\leq \alpha$. If $\sup A$ is no limit ordinal, then $\sup A<\alpha$, ok. If $\sup A$ is a limit ordinal, then the condition $|\alpha|>\gamma$ is needed to show $\sup A<\alpha$.
On the other hand, if $\sup A<\alpha$, can't I conclude that $|\sup A|<|\alpha|$ if $\alpha$ and $\sup A$ are limit ordinals? Maybe not. What is a counterexample?
Consider $\gamma=\aleph_\omega$ and $A=\{\aleph_n: n\in\mathbb N\}$ then $\sup(A)=\gamma$ but $|A|=\omega<\gamma$.