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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).

Martin Sleziak
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jg1896
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  • I'm not sure but I think the Keisler–Shelah theorem never used the full GCH that $2^\kappa=\kappa^+$ for every infinite cardinal $\kappa$, only that there are arbitrarily large $\kappa$ for which it holds. – bof Nov 28 '23 at 21:21
  • In model theory, I believe GCH is used to prove the existence of homogeneous universal structures. I can't quote the exact results offhand. I think a basic paper on this subject is Jónsson's Homogeneous universal relational systems. But this should be in Chang & Keisler. – bof Nov 28 '23 at 21:27
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    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. – Joel David Hamkins Nov 28 '23 at 22:01
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    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 https://mathoverflow.net/a/70435/1946. – Joel David Hamkins Nov 28 '23 at 22:01
  • @JoelDavidHamkins sorry, I should have said that Steinitz was the first non-set theoretical use of it, of course. I will edit my quesiton. – jg1896 Nov 28 '23 at 22:16
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    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. – Andrés E. Caicedo Nov 29 '23 at 04:41
  • @jg1896, Do you mean GCH specifically as opposed to CH? – Mikhail Katz Nov 29 '23 at 10:07
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    @MikhailKatz Just CH is fine. – jg1896 Nov 29 '23 at 10:10
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    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. – Monroe Eskew Nov 29 '23 at 10:47
  • Along the lines of Andrés E. Caicedo's comment, Mary Ellen Rudin proved that CH implies that $\mathbb{R}^\infty$ with the box topology is paracompact. I think it's still open whether CH is needed here. – Timothy Chow Nov 30 '23 at 13:08

2 Answers2

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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.

Dave Benson
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    When I was reading Weibel's book on homological algebra and saw this, I must confess, I was pretty surprised. – jg1896 Nov 28 '23 at 21:34
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    Surprise is generally what I seek in mathematics. But sometimes, I have to resort to astonishment. – Dave Benson Nov 28 '23 at 21:38
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    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. – Denis T Nov 29 '23 at 01:41
  • @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 $n$? – Timothy Chow Nov 30 '23 at 13:12
  • This example is also mentioned in another MO question about CH. – Timothy Chow Nov 30 '23 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.)

Mikhail Katz
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    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. – Joel David Hamkins Nov 29 '23 at 11:02
  • Very interesting point. Something is known about the reverse implication, i. e., does the hyperreal field being unique up to isomorphism implies CH? – jg1896 Nov 29 '23 at 11:05
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    Yes. See this answer. @jg1896 – Mikhail Katz Nov 29 '23 at 11:08
  • @JoelDavidHamkins, I think we have been over this before, but at any rate I think the axiomatic approach to NSA shows that your objection is not that relevant to actual applications of infinitesimals. Indeed, in axiomatic NSA one works directly in $\mathbb R$ rather than any extension thereof, and therefore there couldn't be any issue of "lack of categoricity" as you put it. Meanwhile, many proofs in NSA (certainly all proofs at the level of classical analysis) are easily rephrased in one of the axiomatic frameworks, where categoricity prevails. ... – Mikhail Katz Nov 29 '23 at 11:18
  • ... It has been thought by some that certain accomplishments of NSA, such as Loeb measures and nonstandard hulls, cannot be handled in axiomatic approaches. However, this publication shows otherwise. – Mikhail Katz Nov 29 '23 at 11:18
  • Oh, I don't dispute any of that, and my comment was intended as supportive rather than critical. My point was that "the hyperreal field" is not meaningful in ZFC (not even in second-order logic), since there are several non-isomorphic candidates, and I believe that this is an important part of why it is less studied. With CH, however, there is a canonical model, which would seem to address this concern. (Similarly NSA does not have the same categoricity as is true in ZFC_2.) – Joel David Hamkins Nov 29 '23 at 11:28
  • @JoelDavidHamkins, What you seem to be saying is that hyperreal fields are studied less as set-theoretic objects in themselves because of non-uniqueness. However, this question seems to deal mainly with applications (of CH) in ordinary mathematics, where NSA has much to recommend itself. The reasons NSA is not used more widely are addressed in an article I just sent you via email :-) – Mikhail Katz Nov 29 '23 at 12:09
  • I am trying to describe something more like this: https://twitter.com/JDHamkins/status/1729860011256320012. – Joel David Hamkins Nov 29 '23 at 13:51
  • @JoelDavidHamkins, I don't have an account on $\mathbb X$ so all I see is the following comment of yours: "I should like to sketch an imaginary mathematical history, an alternative history by which the continuum hypothesis (CH) might have come naturally to be seen as a core axiom of set theory and one furthermore necessary for ordinary mathematics." If there is an associated article, I would be interested in seeing it. – Mikhail Katz Nov 29 '23 at 14:00
  • It is a multi-tweet thread, currently only on twitter, but I am developing the idea for other work. – Joel David Hamkins Nov 29 '23 at 14:10
  • @JoelDavidHamkins, I would like to propose a related alternative history. The starting point is that du Bois-Reymond had a concept similar to saturation. My imaginary mathematical history would have him, Frege, and Peano developing axiomatics for infinitesimal analysis (instead of attacking infinitesimals). – Mikhail Katz Nov 29 '23 at 14:16
  • Here is the unrolled version: https://threadreaderapp.com/thread/1729860011256320012.html – Joel David Hamkins Nov 29 '23 at 16:04
  • @JoelDavidHamkins: You wrote: "Let us imagine that in the early days of calculus, the theory was founded upon infinitesimals, but in a clear manner that identified the fundamental principles relating the real field ℝ to the hyperreals ℝ, identifying the transfer principle and saturation." I think it is more likely that in the early days of the calculus, the theory was founded on an assignable vs inassignable distinction within* the ordinary system of real numbers. That's anyway the situation in Leibniz. – Mikhail Katz Nov 29 '23 at 16:09
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    @JoelDavidHamkins: A nice text! As I recall, Martin's axiom may be sufficient for the uniqueness, but I have to check. – Mikhail Katz Nov 29 '23 at 16:11
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    @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. – Philip Ehrlich Nov 29 '23 at 19:03
  • Hi @Philip, nice to hear from you. Incidentally, would GCH be sufficient to ensure the isomorphism of theorem 20 in your "absolute" paper? – Mikhail Katz Nov 30 '23 at 09:50
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    @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. – Philip Ehrlich Nov 30 '23 at 14:32