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If I wanted to write, there exists an integer $n$ then in symbols this would be written as $$\exists n\in\mathbb{Z}.$$ If I wanted to write, there exists only one integer $n$ then in symbols would be written as $$\exists!n\in\mathbb{Z}.$$ Sometimes this could be mistaken as there exists $!n$ but if so that would be written as $$\exists(!n)\ \lor \ \exists\{!n\}.$$ But what if I wanted to write there exists infinitely many integers $n$. I can’t write $$\exists \text{ infinite } n \in\mathbb{Z}$$ or perhaps $$\exists\text{ inf }n\in\mathbb{Z}$$ can I? If not, is there some mathematical notation to denote this? I thought of something like $$\operatorname*{\text{Exi}}\limits_{n\to\infty}\tag{inspired by $\lim_{n\to\infty}n$}n\in\mathbb{Z}$$ but before I make something up, I want to know for sure if there does not exist a notation of this sort yet. This question came to me after looking at this post.

Thank you in advance.

Mr Pie
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You mean something like : $$ \exists A \in \mathcal{P}(\mathbb{Z}), \ |A|=\infty, \quad \forall n \in A, \ \cdots $$ (or card$(A)=\infty$)?

Netchaiev
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  • If you can so easily write $|A| = \infty$, why bother with $A$ at all? Just $|\Bbb Z| = \infty$ would work fine, no? And this entire question would be moot. – Arthur Feb 02 '18 at 09:14
  • @Arthur : you are totally right about $\subset$, it was a (corrected) misprint, thank you! For the second part, we want to say that "there exists infinitely many integer", here the set $A$, not all of them ; but I'll add a part to make it more clear. – Netchaiev Feb 02 '18 at 09:18
  • What is $\text{card}(A)$? I assume that is read as the cardinal of $A$, right? – Mr Pie Feb 02 '18 at 09:31
  • @user477343: yes it is and a quite used notation. – Netchaiev Feb 02 '18 at 09:32
  • @Netchaiev I use $n(A)$ or $#A$ to denote that (particularly the latter when I have already used $n$ as a variable) but I have never seen $\text{card}(A)$ before. Thanks for that :)…… And by the way, I would prefer something like $\text{crd}(A)$ or something (just giving my opinion; of the notation). – Mr Pie Feb 02 '18 at 09:33
  • @user477343: Ok :) I've seen already $# A$. – Netchaiev Feb 02 '18 at 09:42
  • Also, from the duplicate answer, do you know how we would denote there exists at least one integer $x$ or something along those lines? Perhaps, $$\bigg(\operatorname{\exists}^{\geqslant 1}x \ \lor \ \operatorname{\exists}_{\geqslant 1}^\infty x\bigg)\in\mathbb{Z}$$ or something like that? – Mr Pie Feb 02 '18 at 09:42