Notation for the set of all finite subsets of $\mathbb{N}$
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 ..."
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 $\mathcal P_{\mathrm{fin}}(\mathbb N)$.