Solution
= Solution
Map $S\subseteq\omega$ to its <indicator function> $\mathbf1_S:\omega\to\{0,1\}$. Its inverse maps a <binary sequence> $u$ to $\{n\in\omega:u(n)=1\}$. These maps are mutually inverse, so
$$
\boxed{|{}^{\omega}2|=|\mathcal P(\omega)|=2^{\aleph_0}.}
$$
Here the left superscript denotes the <set> of <functions> from $\omega$ into $2$. It is a space of <binary sequences>, not an <ordinal> square.