How to write “there exists an infinite number of”? We all know that ∃ means “there exists” and ∃! means “there exists exactly one”. Is there a similar notation for existence of an infinite number of examples?
 A: I already suggested $\exists^\infty x P(x)$, let me try to collect some relevant links. I agree that perhaps this notation is not standard, but I have seen it often enough, and it seems a natural choice of notation to abbreviate the statement that there are infinitely many $x$ that satisfy property $P(x)$. 
It could be easily explained and agreed upon, at the beginning of a paper or an exposition, and then use it without danger of confusion. 
A related MSE question (the most voted up answer by François G. Dorais suggests $\forall^\infty$ "for all but finitely many", and $\exists^\infty$ "there are infinitely many"). 
Symbols for Quantifiers Other Than $\forall$ and $\exists$ 
A related MO question https://mathoverflow.net/questions/74739/almost-all-quantifier (the discussion there is related to semantics and logic, rather than merely the notation). 
Discussion at wikipedia (e.g. list things like $\exists^\mathrm{many}$, in the context of probability spaces,apart from formal logic discussion). 
http://en.wikipedia.org/wiki/Quantifier_%28logic%29
Stanford Encyclopedia of Philosophy, Generalized Quantifiers (does not specifically advocate the use of $\exists^\infty$ but the discussion is related): 
http://plato.stanford.edu/entries/generalized-quantifiers/
A paper by M.Weese, uses $Q_0$ for "there exist infinitely many"
http://www.jstor.org/discover/10.2307/2040997?uid=3739256&uid=2129&uid=2&uid=70&uid=4&sid=21104634537081
I admit I didn't find as much as I wish online, to confirm that the notation $\exists^\infty$ is standard (and some of what I found was related specifically to logic, rather than to "everyday" mathematics), but I know I did not invent it, and I have seen it used in papers. 
