The continuum hypothesis is known to be independent (neither provable nor disprovable) within the ZFC axioms. But as I understand it, mathematical realists (e.g. Platonists) believe that there is a single "correct" model of mathematics that corresponds to the real world, and therefore that every well-formed mathematical proposition is either "actually true" or "actually false", regardless of whether it can be proven in any particular axiom system.
The continuum hypothesis is perhaps the simplest and most intuitive claim known to be undecidable within ZFC. Do most mathematical realists believe it to be true or false?