I've learned that any Borel set in $\mathbb R^n$ is either countable or of cardinality continuum from Springer's Problems and Theorems in Classical Set Theory. I wonder if there exists some more elementary proofs for the specific case of $G_\delta$ sets in $\mathbb R$.
Below is my trying on this but got stuck.
Suppose $A=\cap_{i=0}^\infty V_i$, where each $V_i$ is open and for each $i\in\mathbb N$, $V_{i+1}\subseteq V_i$. If $A$ contains an interval, then its cardinality equals that of the continuum. Suppose that no interval is contained in $A$. Let $V_i=\cup_{j=0}^\infty J_{i,j}$, where $J_{i,j}$ are disjoint open intervals (and without losing generality we can assume that all with finite length), and for each $i>0, j\in\mathbb N$, there is a unique $J_{i-1,j_{i-1}}$ such that $J_{i,j}\subseteq J_{i-1,j_{i-1}}$. Hence $\forall a\in A$, $\{a\}=\cap_{i=0}^\infty J_{i,j_i}$ and the correspondence between $a$ and the sequence of naturals $\{j_i\}_{i=0}^\infty$ is one-to-one. But I failed to find an one-to-one mapping from the set of all the possible sequences to a subset of $\mathbb N$ or $\mathbb Q$.