By GCH I mean the Generalized Continuum Hypothesis. Let me give some context before presenting my question.

When the axiom of choice was introduced by Zermelo in his 1904 proof of Well-Ordering Theorem, it generated a lot of controversy, and it had many critics. Nowadays, it is used almost freely by mathematicians. Of course we have Gödel's theorem that AC is equiconsistent with ZF (and hence the axiom is 'safe'), but I think the main reason for it being so widely accepted is because it is so useful in proving interesting facts.

To the best of my knowledge, the first time it was used freely (outside set theory) is in the landmark paper by Steinitz on the theory of fields. As the situation gradually evolved, many other interesting facts were discovered using it. In fact, some extremely important results are actually equivalent to it, such as Krull's maximal ideal theorem or Tychonoff's theorem.

My question is this: what are some important theorems in mathematics (outside set theory) that follow from assuming GCH?

I have read parts of Chang and Keisler famous book on Model Theory, and if I remember correctly it used GCH in some proofs.

As an example, we have the remarkable Keisler-Shelah theorem, that says that given first-order language $\mathcal{L}$, two $\mathcal{L}$-structures $\mathcal{M}$ and $\mathcal{N}$ are elementarily equivalent if and only if there is a set $\mathcal{I}$ and a non-principal ultrafilter $\mathcal{U}$ on $\mathcal{I}$ such that $\prod_{i \in \mathcal{I}} \mathcal{M}/\mathcal{U}$ and $\prod_{i \in \mathcal{I}} \mathcal{N}/\mathcal{U}$ are isomorphic. It was first proved assuming GCH, and later a proof not assuming it was obtained.

So, at least in Model Theory, during some period, assuming GCH was useful for obtaining some interesting results.

And what about nowadays? Are there important results whose only known proof assumes GCH? I am particularly interested in results obtained from GCH that mathematical intuition says that ought to be true. I know this last part is inherently subjective, but to give an example of what I mean, there are some computations of global dimension of products of fields which are determined by GCH, but I don't see that these results 'ought to be true' (this is just my impression, of course).

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