How is $ \mathbb Q$ related to $\mathbb R\;$? $\mathbb{R}$ is a set. By one construction, its elements are precisely those sets called "cuts".
$\mathbb{Q}$ is a set. None of its elements are "cuts."
Thus, $\mathbb{Q} \cap \mathbb{R}$ must intersect on the empty set. However, most mathematicians would find such logic overly pedantic (I think).
How can I reconcile this?

There exists an injective relation from $\mathbb Q$ to $\mathbb R$
$$q\mapsto \{x \in \mathbb Q| x< q\}$$
and typically, we speak about the image of $\mathbb Q$ under that injection, not about $\mathbb Q$ itself, so we say that $\mathbb Q\subseteq\mathbb R$.
