Assume ZFC. Let $B\subseteq\mathbb R$ be a set that is not Borel-measurable. Clearly, $B$ must be uncountable, since countable sets are always Borel being a countable union of measurable singletons.
Question: can one conclude that $B$ necessarily has the cardinality of the continuum without assuming either the continuum hypothesis or the negation thereof?
A possibly related result is that any $\sigma$-algebra that contains infinitely many sets must necessarily have at least the cardinality of the continuum. This result is independent of the continuum hypothesis.