= Solution
The <Freiling axiom of symmetry> is
$$
\boxed{\forall f:\mathbb R\to[\mathbb R]^{<\omega_1}\quad
\exists x,y\in\mathbb R\quad
x\notin f(y)\ \land\ y\notin f(x).}
$$
The values of $f$ are countable <subsets> of the <real numbers>, including finite ones. One may require $x\ne y$ without changing the axiom: apply the displayed version to $x\mapsto f(x)\cup\{x\}$.
The <Generalized continuum hypothesis> states
$$
\boxed{\forall\text{ infinite cardinals }\kappa,\quad2^\kappa=\kappa^+.}
$$
Here $\kappa^+$ is the <successor cardinal>; equivalently $2^{\aleph_\alpha}=\aleph_{\alpha+1}$ for every <ordinal> $\alpha$.
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