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.
On
"$P(x)$ holds for all but finitely many $x$" means "$P(n)$ holds for all except finitely many $x$" here. For instance, if $(x_{n})$ converges to $x$ for some $x$, then given any $r > 0$ we can say "$|x_{n}-x| < r$ for all but finitely many $n$". You see why? The definition only requires $|x_{n}-x| < r$ from some $n$ on, meaning there could be finitely many $n$ such that $|x_{n}-x| \geq r$.
The negation of "$P(x)$ is true for all but finitely many $x$" is thus "$P(x)$ is false for infinitely many $x$".
The phrase "$P(x)$ is true for all but finitely many $x$ in some given finite set" is not informative.
Regardless of what $X$ is, "$P(x)$ holds for all but finitely many $x\in X$" means that the set $\{x\in X:\neg P(x)\}$ is finite. If $X$ is finite, then $\{x\in X:\neg P(x)\}$ is always finite, since it is a subset of $X$. So if $X$ is finite, every property holds for all but finitely many $x\in X$.