It is known that closed sets of $\mathbb{R}$ satisfies continuum hypothesis, that is, every closed subset of $\mathbb{R}$ is either countable or of the cardinality of the continuum.
Is the cardinality of uncountable $G_{\delta}$ set of $\mathbb{R}$ equals the cardinality of the continuum?
Yes. In fact the cardinality of every Borel set is countable or size continuum.
There is an easy proof for $G_\delta$. Every $G_\delta$ set is completely metrizable, and therefore either countable or contains a perfect set (which has a copy of a Cantor space).
(See this answer for the former fact, and the Cantor-Bendixson derivative, and related theorems for the latter)
