 Solution

The Gödel constructible universe theorem gives $\mathrm{Con}(\mathrm{ZFC})\Rightarrow \mathrm{Con}(\mathrm{ZFC}+\mathrm{CH})$. The forcing independence theorem for the Continuum hypothesis gives $\mathrm{Con}(\mathrm{ZFC})\Rightarrow \mathrm{Con}(\mathrm{ZFC}+\neg \mathrm{CH})$. For example, the latter can be obtained by first passing to the constructible universe and then adding sufficiently many Cohen reals. By the equivalence in part (b), these are respectively models of the negation and affirmation of the free-pair assertion. Thus if ZFC is consistent, the assertion is independent of ZFC. The consistency qualification is essential: an inconsistent theory proves every sentence. These are syntactic relative-consistency implications, not a claim that bare consistency supplies a countable transitive model.