TITLE: Meaning of "occurs for infinitely many" and "all but finitely many"
QUESTION [13 upvotes]: Suppose $P(n)$ is certain statement. I hear all the time that my teachers say 
$$ P(n) \; \; \text{occurs for infinitely many} \; \; \;n $$
$$ P(n) \; \; \text{for all but finitely many} \; \; n $$
MY question: What are the precise definitions of this statements ?

REPLY [4 votes]: The first statement ($P(n) \; \; \text{occurs for infinitely many} \; \; \;n$) means that there is an infinite number of ways to satisfy the proposition $P(n)$. Just like Michael says, in $\mathbb(n)$ let $P(n)$ : n is even, then there is an infinite number of $n \in\mathbb N$ that satisfies $P(n)$ (because there are infinitely many even numbers).
The second statement ($P(n) \; \; \text{for all but finitely many} \; \; n$) means that, although there is an infinite number of ways of satisfying a condition $P(n)$, there is a finite number of exceptions. For example, in $\mathbb N$ let $P(n)$ : n > 5. There is a inifinite number of $n \in\mathbb N$ that satisfies $P(n)$ (there are infinitely many natural numbers greater than 5), but there is a finite number of exceptions, in our case, the numbers: 0, 1, 2, 3, 4 and 5 are those exceptions.