Is there in literature a descriptive abbreviation phrase for "for infinitely many $n$"? Let $P(n)$ be a property for all $n \geq 1$. For the phrase "there is some $N \geq 1$ such that $P(n)$ holds for all $n \geq N$" there are some suggestive, convenient abbreviations such as "$P(n)$ holds for large $n$" or "$P(n)$ holds eventually" and so on.
I wonder if there is in literature a like abbreviation for "$P(n)$ holds for infinitely many $n$"? I am aware that in probability theory some authors would write "$P(n)$ holds infinitely often"; but, in my humble opinion, this abbreviation would be not that useful in contexts other than probability theory.
 A: In descriptive set theory and logic one sometimes uses $\exists^\ast_n P(n)$ for "there are infinitely many $n$ such that $P(n)$ holds", and $\forall^\ast_n P(n)$ for "all but finitely many $n$ satisfy $P(n)$".
Then at least we have $\lnot \forall^\ast_n P(n) \leftrightarrow \exists^\ast_n \lnot P(n)$, like for normal quantifiers, e.g.
A: In some contexts (set theory, order theory, point set topology, though probably never in probability) you can say cofinally, or cofinally many, cofinally often. Given a preorder $(A,\preceq)$, a subset $X\subseteq A$ is cofinal in $A \Leftrightarrow$ for every $a\in A$ there is $x\in X$ with $a\preceq x$. A predicate $\varphi(x)$ holds cofinally often, and is true for cofinally many $x$, iff $\{x\in A\mid \varphi(x)\}$ is cofinal in $A$. For $\Bbb N$ with the usual order, this is exactly the same notion as "infinitely often".
"Eventually" is often used in such a setting (vis a vis a preorder) to mean: $eventually_x, \varphi(x) \!\stackrel{def}\iff\!$ there is $a\in A$ such that for all $x\succeq a, \varphi(x)$. In that case, "frequently" can be and has been used rather than "cofinally often".
Note that these two notions are quantifiers, duals of each other:
$$
\text{frequently$_x \varphi(x) \iff \neg$ eventually$_x \neg\, \varphi(x)$. }
$$
In his book General Topology [p.65], Kelley uses the terms "eventually" and "frequently", for special cases of these notions: given a directed preorder $(A,\preceq)$, a function $f\colon A\to B$ is frequently in a subset $Y\subseteq B$ iff $\{x\in A\mid f(x)\in Y\}$ is cofinal in $A$. Similarly for the dual notion "$f$ is eventually in Y". A net $f$ converges to a point $b\in B \!\iff\!$ for every neighborhood $U$ of $b$, $f$ is eventually in $U$.
A: From John L. Kelley's General Topology (available at the Internet Archive), p. 65:

A directed set is a pair $(D,\ge)$ such that $\ge$ directs $D.$ [. . . .] A net $\{S_n,n\in D,\ge\}$ is in a set $A$ iff $S_n\in A$ for all $n$; it is eventually in $A$ iff there is an element $m$ of $D$ such that, if $n\in D$ and $n\ge m,$ then $S_n\in A.$ The net is frequently in $A$ iff for each $m$ in $D$ there is $n$ in $D$ such that $n\ge m$ and $S_n\in A.$

Similarly, I guess you could say "$P(n)$ holds eventually" if $P(n)$ holds for all sufficiently large $n,$ and "$P(n)$ holds frequently" if $P(n)$ holds for infinitely many $n.$
A: I'm aware of only one way how to write this clearly -- in symbols:
$$ \exists_\infty n\in\mathbb{N} : P(n). $$
However, unless you need this a lot you shouldn't use this notation. If you need it a lot, you can introduce it:

We suppose that $P(n)$ holds for infinitely many $n$, i.e., that $\exists_\infty n\in\mathbb{N}:P(n)$.

Some people use: "$P(n)$ holds infinitely many times," but I don't find this one quite nice; also, it's not even shorter. In contexts when no confusion can happen, one can use: "$P(n)$ holds infinitely often." However, be aware that to some, this means that $P(n)$ holds with a positive density.
