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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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    $\begingroup$ You say that the first free use of the axiom of choice is by Steinitz, but why neglect Zermelo's original use? Zermelo proved his well ordering theorem (that every set can be well ordered) originally in an informal set theory, but when people wanted to know exactly what principles were in play, he formulated Zermelo set theory, which included the axiom of choice. It seems clear that he thought of the axiom as expressing a true principle of the nature of sets. $\endgroup$ Nov 28, 2023 at 22:01
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    $\begingroup$ You also say that the main reason AC is accepted is because of its use, in other words extrinsically justified, but many mathematicians find it to have enormous intrinsic support, finding it as Zermelo did to express a basic truth of set theory. Although AC is famously criticized for some unexpected consequences, such as the Banach-Tarski theorem, a more balanced discussion will also note that AC rules out some far more pathological phenemenon, as mentioned in mathoverflow.net/a/70435/1946. $\endgroup$ Nov 28, 2023 at 22:01
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    $\begingroup$ CH is used frequently in various places in analysis. For instance, the theory of automatic continuity on Banach algebras is very different depending on whether you assume CH or strong forcing axioms. $\endgroup$ Nov 29, 2023 at 4:41
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    $\begingroup$ @MikhailKatz Just CH is fine. $\endgroup$
    – jg1896
    Nov 29, 2023 at 10:10
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    $\begingroup$ The question seems to assume GCH and AC have somewhat the same status, both as useful but controversial set theoretic principles, each of whose cases for axiomhood is strengthened by their applications. But AC is nearly universally accepted as a foundational truth with intrinsic appeal, and its usefulness stems from it already having been used, sometimes implicitly, in establishing mainstream mathematics. GCH is not viewed as intrinsically justified. There are other contradictory principles of arguably equal status. It hasn't been implicitly used in the mainstream. $\endgroup$ Nov 29, 2023 at 10:47

2 Answers 2

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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.

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    $\begingroup$ When I was reading Weibel's book on homological algebra and saw this, I must confess, I was pretty surprised. $\endgroup$
    – jg1896
    Nov 28, 2023 at 21:34
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    $\begingroup$ Surprise is generally what I seek in mathematics. But sometimes, I have to resort to astonishment. $\endgroup$ Nov 28, 2023 at 21:38
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    $\begingroup$ This gets less surprising and more substantial when you look closer at the notion of pure-projectivity (which is equivalent to being a direct summand in a sum of finitely presented modules). There's a long series of works by H. Lenzing on pure projective dimensions of finite dimensional algebras, where it's proven that homological properties of $\Bbb Q[\Bbb Z/2 \times \Bbb Z/2]$ depend on CH. Enlarging transcendence degree of the base field and taking slightly more complicated wild algebras, you can state GCH in language of modules over finite dimensional commutative algebras over a field. $\endgroup$
    – Denis T
    Nov 29, 2023 at 1:41
  • $\begingroup$ @DaveBenson I'm a little confused about your corollary regarding $\mathbb{R}(x,y,z)$. In order to apply Osofsky's result as you've stated it, doesn't one need to assume that the cardinality of $\mathbb{R}$ is $\aleph_n$ for some integer $n$? $\endgroup$ Nov 30, 2023 at 13:12
  • $\begingroup$ This example is also mentioned in another MO question about CH. $\endgroup$ Nov 30, 2023 at 13:16
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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.)

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    $\begingroup$ Regarding your last point, I do find it relevant, as well as philosophically significant. Ther reason is that without that uniqueness result, we don't have a referent for "the hyperreal field", and in my view, this lack of categoricity is one of the principal explanations for why use of hyperreals is not adopted more widely. There is no canonical mathematical structure there that we are talking about. But with CH, however, there is. $\endgroup$ Nov 29, 2023 at 11:02
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    $\begingroup$ Yes. See this answer. @jg1896 $\endgroup$ Nov 29, 2023 at 11:08
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    $\begingroup$ @JoelDavidHamkins: A nice text! As I recall, Martin's axiom may be sufficient for the uniqueness, but I have to check. $\endgroup$ Nov 29, 2023 at 16:11
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    $\begingroup$ @JeolDavidHamkins: Joel, Keisler put forth much the same idea you propose in your "unrolled version" on pp. 228-230 of his paper "The Hyperreal Line” in "Real Numbers, Generalizations of the Reals, and Theories of Continua", ed. by Philip Ehrlich, Kluwer Academic Pub. 1994. $\endgroup$ Nov 29, 2023 at 19:03
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    $\begingroup$ @Mikhail Katz: Hi Mikhail. Under GCH, one gets uniqueness of saturated hypperreal number systems as characterized in that paper for sufficiently large $\kappa$. One needs to have $\kappa > \aleph_2$ because the isomorphism theorem for saturated models requires that $\kappa$ is greater than the cardinality of the vocabulary, and the set $F$ of real predicates has cardinality $\aleph_2$ under GCH. However, I suspect without global choice, one cannot get uniquesness for the class structure from GCH alone. $\endgroup$ Nov 30, 2023 at 14:32

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