= Solution
Work over a ground model of <ZFC>+<Generalized continuum hypothesis> and let $\lambda=\aleph_3$. Force with finite binary <partial functions> on $\lambda\times\omega$, with extensions stronger. The <Delta-system lemma> thins any uncountable family of finite domains to an uncountable family with one common root. Only finitely many binary assignments on that root occur, so two conditions agree there and their union is a common extension. Hence the <forcing> has the <countable chain condition for forcing> and preserves <cardinals> and <cofinalities>.
The generic union yields $\lambda$ distinct reals. Totality at each coordinate is dense, and for two different coordinates it is dense to assign different values at some unused natural-number position. Thus the extension satisfies $2^{\aleph_0}\ge\lambda$.
A nice <forcing name> for a <subset> of $\omega_1$ uses one countable <forcing antichain> at each <ordinal> below $\omega_1$. Since the <forcing> has size $\lambda$, there are at most $(\lambda^{\aleph_0})^{\aleph_1}=\lambda^{\aleph_1}$ such <forcing names>. Ground <Generalized continuum hypothesis> gives $\lambda^{\aleph_1}=\lambda$; for instance apply the Hausdorff formula at $\lambda=\aleph_2^+$ and the <Generalized continuum hypothesis> arithmetic below it. Therefore $2^{\aleph_1}\le\lambda$ in the extension. Combining the bounds gives
$$
\boxed{2^{\aleph_0}=2^{\aleph_1}=\aleph_3.}
$$
The <ordinal> and <cardinal> $\aleph_3$ is the same in both models by the <chain in a partial order> condition. The <forcing theorem> formalizes this construction as the requested relative-consistency implication. A countable transitive ground model is a convenient presentation, not an additional consequence silently derived from mere consistency. This is the <Cohen forcing two-level continuum plateau>.
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