# Continuum Hypothesis $\iff ?$?

I have read that CH cannot be proved nor disproved within ZFC, and I was wondering:

1. Which (If any) branches/fields of Mathematics are built upon CH being true?
2. Are there any subjects built upon It's falsity?
• If I recall correctly, there was a related thread recently. math.stackexchange.com/questions/648550/… – Asaf Karagila Feb 13 '14 at 18:52
• @AsafKaragila CH doesn't deserve a tag? :) – user76568 Feb 13 '14 at 18:53
• I don't find it a substantial enough topic to merit a tag on MSE. So no. Questions about CH are generally labeled under set-theory and/or cardinals, and the string "continuum hypothesis" is easily searchable. – Asaf Karagila Feb 13 '14 at 18:54
• In any case, using a single tag which is new guarantees almost zero exposure of the question. If anything, I helped ensuring that other people will see it. – Asaf Karagila Feb 13 '14 at 18:56
• @AsafKaragila I see. Thanks then :) – user76568 Feb 13 '14 at 18:59

In operator theory, and in particular $C^*$-algebra, there are some statements regarding automorphisms of the Calkin algebra whose truth value depends on $\sf CH$.

The global dimension of a countable product of fields depends on the value of $\sf CH$ as well, and the Kaplansky conjecture for Banach algebras also suffers from the same fate as those before it.

While I don't know of any branches or fields of mathematics where the truth value of $\sf CH$ plays such a significant role (the above examples are minor results in analysis and algebra), I doubt there are any such fields. For two main reasons:

1. The independence phenomenon is relatively new, and while it began propagating outside of set theory and into mainstream mathematics in the 1970's (Shelah and the Whitehead problem), it usually requires some pre-existing interest. This means that it is usually applied to a problem of the following formulation:

Some property holds trivially for finitely generated objects; with some effort we can prove it for countably generated objects. Does it hold for any object?

In that case it is often the case that cardinal arithmetic will have something to say, and sometimes it turns out that this something is independence.

However it is only in the past few decades that mathematics is leaving the shell of finitely/countably generated objects. So there's still some distance before $\sf CH$ will become a household assumption (or its negation). Even then, it is often the case that we need more than just $\sf CH$ and its negation (e.g. $\lozenge_{\omega_1}$ or $\sf MA_{\aleph_1}+\lnot CH$) in order to prove the results, and $\sf CH$ by itself is insufficient (as Shelah proved with respect to the Whitehead problem).

2. Set theorists, while interested in the application and independence of general mathematical statements from set theoretical axioms, are unlikely to sit and develop entire fields or branches by themselves which depend solely on $\sf CH$. Set theorists would be interested in general independence, or generalized statements, not just $\sf CH$ itself.

On the other hand, most general mathematicians that I have seen and spoke with would either be uninterested in the set theoretical assumptions (because they are unneeded for the "interesting part" of their field) or would be interested in the assumptions and independence, but will actively pursue consequences of both assuming $\sf CH$ and assuming its negation, so there it seems unlikely that there will be a field which solely depends on $\sf CH$ itself.

With the above been said, in set theory the theory of cardinal characteristics of the continuum trivializes completely when assuming $\sf CH$, so it is usually studied under its negation. But even then many of the constructions begin with a model of $\sf CH$ and by using forcing prove consistency of statements.

• There are quite a few results in real analysis and classical point set theory that require the Continuum Hypothesis, as Sierpinski showed in his 1934 book Hypothèse du continu. – Dave L. Renfro Feb 13 '14 at 19:57
• Dave, thank you for the reference. As I wrote, there are probably such results in any field which deals with uncountable objects. But there is no field whose foundation lies on $\sf CH$, or the failure thereof. – Asaf Karagila Feb 13 '14 at 20:01
• @AsafKaragila Well, not a field, but the theory of discontinuous homomorphisms between Banach algebras (of which there is much to say) is an example; Kaplansky's conjecture is just the beginning of the story. The work on the Calkin algebra is much more recent, but it is shaping up similarly. So: A large body of work within functional analysis is explicitly dependent on $\mathsf{CH}$, and this is not a collection of isolated or made up examples but rather a coherent theory. – Andrés E. Caicedo Feb 13 '14 at 21:07
• @Andres: Interesting. From discussing this with a BGU classmate who is now an excellent Ph.D. student in the field of $C^*$-algebra, I had the impression that the whole set theoretic applications, while major within set theory, are considered a niche (an expanding one, but a niche nonetheless) by non-set theorists of the field. So while there is a theory to be developed here, it seems that the theory is not "Assume $\sf CH$" but rather "Explore the implications of $\sf CH$, and its negation". Both which, to my reading of the question, are not quite the examples the OP were looking for. – Asaf Karagila Feb 13 '14 at 21:21
• I would not know on the Calkin algebra; again, this is very recent, so opinions may not be settled yet. Not so for the theory of discontinuous homomorphisms. The question of when two algebra norms are equivalent was studied classically, and once we have two non-equivalent ones, one can use this as a starting point for a whole series of results. It is not a matter of constructing pathological examples, and it is very much functional analysis rather than set theory guiding the developments. Dales huge book treats this in detail, and there are quite a few other references. – Andrés E. Caicedo Feb 13 '14 at 21:28