Notation for the set of all finite subsets of $\mathbb{N}$ Is there a "standard" notation to denote the set of all finite subsets of $\mathbb{N}$?
(or any set, not just $\mathbb{N}$)
Thanks
 A: Several possible notations for $\{A\subseteq\omega\mid |A|<\omega\}$:


*

*$[\omega]^{<\omega}$

*$P_\omega(\omega)$

*$\operatorname{Fin}(\omega)$


Where, of course, $\omega=\mathbb N$.
And as usual my advice on the matter: When in doubt, open with "We denote by [the chosen notation here] the set ..."
A: You can find various notations, as mentioned in coments. (I doubt there is some generally accepted notation.)


*

*You can find $[\omega]^{<\omega}$, e.g. here, which can be considered as a special case of $[A]^{<\kappa}$ - which denotes all subsets of $A$ of cardinality less then $\kappa$, see e.g. p.18 of the same book. In your case you could use $[\mathbb N]^{<\omega}$.

*You can find $\mathrm{Fin}$, e.g. here and here

*You can find $\mathbb N^{[<\infty]}$, e.g. here.

*Hindman and Strauss use $\mathcal P_f(\mathbb N)$ in this book, which is similar to Qiaochu's suggestion $\mathcal P_{\mathrm{fin}}(\mathbb N)$.
A: In addition to the answers already given, this is just
$$\bigcup_{k \in \mathbb{N}} \binom{\mathbb{N}}{k}$$
where $\binom{S}{k}$ is the set of $k$-subsets of $S$.
