If $(P,\leq)$ is a poset and $S\subseteq P$ we let $$\uparrow S = \{p\in P: p\geq s\text{ for some }s\in S\}.$$

Let $([\omega]^\omega,\subseteq)$ denote the collection of infinite subsets of $\omega$, ordered by set inclusion. If $S\subseteq [\omega]^\omega$ has the property that $\uparrow S = [\omega]^\omega$, does this imply $|S|=2^{\aleph_0}$?