= Freiling theorem
{c}
{title2=$A_{<\omega_1}(\mathbb R)\Longleftrightarrow\neg\mathrm{CH}$}
= Countable-valued free-pair criterion
{synonym}
In <ZFC>, the <Freiling axiom of symmetry> is equivalent to failure of the <Continuum hypothesis>. Under the hypothesis, countable initial segments of a well-order of the reals violate symmetry. When the continuum exceeds $\aleph_1$, the union of the countable values on an $\aleph_1$-sized set leaves a real outside; avoiding that real's countable value gives the two witnesses.
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