'All but finite' and 'infinitely many' There are two phrases in Mathematics: 'All but finite' and 'infinitely many'. I have some confusion between them.
Suppose $P(n)$ is a statement. If we say that $P(n)$ is true for all but finite number of values of $n$, it means it is true for all values of $n$ except a finite number of values of $n$.
If $P(n)$ is true for infinitely many number of values of $n$, so does it mean that it is true for all values of $n$ except infinite values of $n$?
 A: If something is true of all positive even numbers $2,4,6,8,10,12,\ldots$, and not true of odd numbers, then it is true of infinitely many of the positive integers $1,2,3,4,5,\ldots$.
But it is not that case that it's true of all except finitely many, since the set of numbers of which it is not true --- the odd numbers --- is not finite.
A: You might consider thinking of it like this.
Suppose $P$ is true for "all but finitely many objects". Then you can actually list those objects: $x_1, x_2, x_3, \dots, x_n$ all fail to have property $P$.
Now you might ask, does $P$ hold for infinitely many objects? You can probably see that it will only be true if the domain we're talking about is infinite: If $P$ is a property of the integers, then finitely many "bad" cases gives us infinitely many "good" cases. 
But we could also be working over a finite domain in some special situations. Suppose we're studying Woozzle integers, a special subset of the integers just discovered. We want to know how many Woozzle integers have property $P$. But the weird thing is, no one knows how many Woozzle integers there are yet. So while someone could reasonably prove that $P$ holds for all but finitely many Woozzle integers, you could not then conclude that it holds for infinitely many of them, since it's not clear there even are infinitely many.
A: Indeed, given statements $P(n)$ hold for all but finitely many $n \in \mathbb{N}$ means that there is some $N \in \mathbb{N}$ such that $n > N$ implies $P(n)$ holds; the statements $P(n)$ hold for infinitely many $n \in \mathbb{N}$ means that for every $N \geq 1$ there is some $n > N$ such that $P(N)$ holds. 
