"All but finitely many..." in the finite case Maybe this is a stupid question, but I can't figure it out, since the wording makes it very confusing to me. I've been mulling over the meaning of "all but finitely many" and its negation. There are plenty of answers to what that means. But what does this mean if some property $P$ is defined on a finite set $X$? To say that "$P(x)$ holds for all but finitely many $x\in X$" means what? Thank you in advance.

Regardless of what $X$ is, "$P(x)$ holds for all but finitely many $x\in X$" means that the set $\\{x\in X:\
eg P(x)\\}$ is finite. If $X$ is finite, then $\\{x\in X:\
eg P(x)\\}$ is always finite, since it is a subset of $X$. So if $X$ is finite, every property holds for all but finitely many $x\in X$.