Is the cardinality of uncountable $G_{\delta}$ set of $\mathbb{R}$ equals the cardinality of the continuum?

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?

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