Uniqueness results that follow from CH

Question

Recently, Joel David Hamkins presented a historical thought experiment that shows that CH could have been adopted as an axiom if we had been using the hyperreal field $\mathbb{R}^*$ instead of $\mathbb{R}$ in calculus, since we'd like to have a categoricity result for $\mathbb{R}^*$ in this alternative history.

Regardless of the validity of this argument, which I am not planning to discuss here, I thought it would be nice to have a collection of uniqueness results that follow from CH, not to claim that CH should be an axiom but simply to have such a list for its own sake. There seems to be no similar question on MO as far as I can tell. Therefore, my big-list question is:

What are some uniqueness results (whose negations are believed/known to be relatively consistent with ZFC) that follow from CH?

Answer 1

An example would be the uniqueness of the asymptotic cone for uniform lattices in $SL_m(\mathbb R)$. Under $\neg CH$, these admit $2^{2^{\aleph_0}}$ non-homeomorphic asymptotic cones.

See Asymptotic cones of finitely presented groups, by Linus Kramer, Saharon Shelah, Katrin Tent and Simon Thomas.

Answer 2

Under CH, we have saturated models of size continuum of any consistent first-order theory in a countable language, and for a complete theory these are unique by the back-and-forth method.

(In my paper, I had pointed out that the CH is actually equivalent to the assertion that there is a unique smallest countably saturated real-closed field.)

Answer 3

Assuming $\textsf{CH}$, a lot of natural fourth-order functionals are computationally equivalent (Kleene's S1-S9) to $\exists^3$. These equivalences do not seem to go through without the former. ([1])

[1] Dag Normann and Sam Sanders, The computational properties of the Axiom of Choice, Journal of Logic and Computation, 2022.

Answer 4

Here's one about my favorite topological space, the Stone–Čech remainder of the natural numbers, denoted $\mathbb N^*$ or $\omega^*$. The characterization is due to Parovicenko.

Assuming CH, $\mathbb N^*$ is the unique compact Hausdorff zero-dimensional space such that (1) there is a basis of size $\mathfrak c$, (2) every nonempty $G_\delta$ has infinite interior, and (3) it is an $F$-space (disjoint open $F_\sigma$'s have disjoint closures).

Let me note that $\mathbb N^*$ always has these properties. But only under CH is it the only space with these properties. In other words, this description of $\mathbb N^*$ characterizes it uniquely if and only if CH holds. This was proved by van Douwen and van Mill in

van Douwen, Eric K.; van Mill, Jan, Parovicenko’s characterization of (\beta\omega -\omega) implies CH, Proc. Am. Math. Soc. 72, 539-544 (1978). ZBL0393.54004.

Via Stone duality, this can also be used to obtain a description of the Boolean algebra $\mathcal P(\mathbb N)/\mathrm{Fin}$ that characterizes it completely under CH. This is somewhat related to Joel's answer, by the way, because the uniqueness result for $\mathcal P(\mathbb N)/\mathrm{Fin}$ can be proved via saturation.

Answer 5

In line with Joel’s answer, we have all sorts of uniqueness of ultrapower results that follow form CH and whose negations are known to be consistent with ZFC. Since classical model theory is not my area, I’ll let others add those examples. For me, I’ll point out an example in my field: For any separable (or, more generally, of density character less than or equal to continuum, under the $\sigma$-strong topology) II$_1$ factor $M$ (i.e., an infinite-dimensional tracial von Neumann algebra with scalars as its center), its ultrapower $M^\mathcal{U}$ is independent of the choice of free ultrafilter $\mathcal{U}$ on $\mathbb{N}$ iff CH holds.