[proofplan] The proof is a metatheoretic implication chase. Starting from the consistency of $\mathrm{ZFC}$, the assumed relative consistency theorem for $\mathrm{GCH}$ gives the consistency of $\mathrm{ZFC}+\mathrm{GCH}$. The course model-existence metatheorem then converts that consistency statement into the consistency of a theory satisfying $\neg\mathrm{CH}$. Composing the two implications gives the desired relative consistency result. [/proofplan] [step:Introduce the theories and the metatheoretic consistency predicate] Let $\operatorname{Con}(T)$ denote the assertion that the first-order theory $T$ is syntactically consistent, meaning that no contradiction is derivable from the axioms of $T$ in first-order logic. Let $\mathrm{ZFC}$ denote Zermelo-Fraenkel set theory with the [Axiom of Choice](/page/Axiom%20of%20Choice), let $\mathrm{CH}$ denote the Continuum Hypothesis, and let $\mathrm{GCH}$ denote the Generalized Continuum Hypothesis. The theory $\mathrm{ZFC}+\mathrm{GCH}$ is obtained by adjoining $\mathrm{GCH}$ to the axioms of $\mathrm{ZFC}$, and the theory $\mathrm{ZFC}+\neg\mathrm{CH}$ is obtained by adjoining the negation of $\mathrm{CH}$ to the axioms of $\mathrm{ZFC}$. [/step] [step:Use the relative consistency of $\mathrm{GCH}$ over $\mathrm{ZFC}$] Assume $\operatorname{Con}(\mathrm{ZFC})$. By the assumed relative consistency of $\mathrm{GCH}$ over $\mathrm{ZFC}$, we have \begin{align*} \operatorname{Con}(\mathrm{ZFC}+\mathrm{GCH}). \end{align*} [guided] We begin with the hypothesis that $\mathrm{ZFC}$ is consistent: \begin{align*} \operatorname{Con}(\mathrm{ZFC}). \end{align*} The first metatheoretic input says exactly that consistency of $\mathrm{ZFC}$ transfers to consistency of $\mathrm{ZFC}$ together with $\mathrm{GCH}$: \begin{align*} \operatorname{Con}(\mathrm{ZFC}) \implies \operatorname{Con}(\mathrm{ZFC}+\mathrm{GCH}). \end{align*} Applying modus ponens to these two statements gives \begin{align*} \operatorname{Con}(\mathrm{ZFC}+\mathrm{GCH}). \end{align*} This is the intermediate consistency statement needed for the forcing/model-existence part of the argument. [/guided] [/step] [step:Apply the model-existence metatheorem to obtain a model of $\mathrm{ZFC}+\neg\mathrm{CH}$] By the standard model-existence metatheorem used in the course, the consistency of $\mathrm{ZFC}+\mathrm{GCH}$ implies the consistency of $\mathrm{ZFC}+\neg\mathrm{CH}$. Therefore \begin{align*} \operatorname{Con}(\mathrm{ZFC}+\neg\mathrm{CH}). \end{align*} [/step] [step:Compose the implications to conclude the relative consistency of $\neg\mathrm{CH}$] We have shown that the assumption $\operatorname{Con}(\mathrm{ZFC})$ implies \begin{align*} \operatorname{Con}(\mathrm{ZFC}+\mathrm{GCH}) \end{align*} and that this in turn implies \begin{align*} \operatorname{Con}(\mathrm{ZFC}+\neg\mathrm{CH}). \end{align*} Hence, by transitivity of implication, \begin{align*} \operatorname{Con}(\mathrm{ZFC}) \implies \operatorname{Con}(\mathrm{ZFC}+\neg\mathrm{CH}). \end{align*} This is the desired relative consistency statement. [/step]