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?

Solution 1:

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.