= Solution
The <Gödel constructible universe theorem> gives $\operatorname{Con}(\mathrm{ZFC})\Rightarrow\operatorname{Con}(\mathrm{ZFC}+\mathrm{CH})$. The <forcing> independence theorem for the <Continuum hypothesis> gives $\operatorname{Con}(\mathrm{ZFC})\Rightarrow\operatorname{Con}(\mathrm{ZFC}+\neg\mathrm{CH})$. 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 \b[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>.
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