Are there uncountable antichains in $\mathcal{P}(\mathbb{N})$

Enumerate the (countably many) vertices of the infinite rooted binary tree by $\mathbb{N}$. Then each infinite path of this tree will induce an infinite subset of $\mathbb{N}$. Moreover, two such subsets are not comparable with respect to inclusion (and indeed, the intersection of two such subsets will be finite since two distinct infinite paths in this tree have only finitely many common edges.) Now try to count how many infinite paths there are in the infinite rooted binary tree.

It is enough to show there exists an uncountable antichain $\mathcal{A} \subset \mathcal{P}(\mathbf{Q})$.

\begin{align*} \mathcal{A} = \{\ [\alpha-1, \alpha] \cap \mathbf{Q} \mid \alpha \in \mathbf{R} \ \} \end{align*}

satisifies such a condition.

Let $S\subset\mathbb{N}$. Let $$R_S=\{p_s^t:s\in S,t\notin S\}$$, where $p_s$ is the $s^{th}$ prime number.