Full Idea
Many set theorists doubt if the Generalised Continuum Hypothesis must be either true or false; certainly, its bivalence is far from obvious. All the same, almost all set theorists use classical logic in their proofs.
Gist of Idea
Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic
Source
Ian Rumfitt (The Boundary Stones of Thought [2015], 7.2)
Book Reference
Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.196
A Reaction
His point is that classical logic is usually taken to rest on bivalence. He offers the set theorists a helping hand, by defending classical logic without resorting to bivalence.
Related Idea
Idea 18827 If truth-tables specify the connectives, classical logic must rely on Bivalence [Rumfitt]