Is GCH useful in proving theorems?

Question

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).

Answer 1

One surprising (at least to me) area where the GCH enters is the following piece of homological algebra. Let $k$ be a field of cardinality $\aleph_n$. Then Barbara Osofsky proved that the projective dimension of $k(x_1,\dots,x_r)$ as a module over $k[x_1,\dots,x_r]$ is whichever is smaller of $r$ and $n+1$. So for example the projective dimension of $\mathbb{R}(x,y,z)$ as a module over $\mathbb{R}[x,y,z]$ is equal to $2$ if the continuum hypothesis holds and $3$ otherwise. And with more variables, we distinguish between higher cardinalities, thereby bringing in the GCH.

That said, I don't personally have any preference as to whether GCH holds or not; I'd just like to know what the consequences are. Whereas I really do wish to assume that the axiom of choice holds. I'd like to be able to use the well-ordering principle at will, because it's so convenient to do so. For example, I'd like to be able to use the fact that in a commutative ring with unit, every ideal is contained in a maximal ideal.

Answer 2

Dixmier traces are easily constructed in ZFC and there is an extensive literature on the topic. Connes pointed out that such a trace with particularly good properties can be constructed in the assumption of CH. In his article *Brisure de symétrie from 1997 on page 211, Connes argues that his solution is substantial and calculable. Connes showed that a theorem of Mokobodzki from

P. Meyer, Limites médiales, d'après Mokobodzki, Séminaire de Probabilités, VII (Univ. Strasbourg, année universitaire 1971--1972) pp. 198--204. Lecture Notes in Mathematics 321, Springer, Berlin, 1973

provides a limit process (for the Dixmier trace) which is universally measurable. Connes' argument is analyzed in this publication.

(Another example close to my interests: it follows from CH that the hyperreal field $\mathbb R^{\mathbb N}\!/\mathcal U$ is unique up to isomorphism. However, this sounds better than it is, since the isomorphism in question is not internal, and therefore is not that relevant to actual applications of infinitesimals.)