I remember at a certain point early in my mathematical studies learning that the Axiom of Choice is equivalent to the following statement on Cartesian products:

If $\{ X_i \}_{i \in I}$ is any collection of nonempty sets indexed by an index set $I$, then $\prod_{i \in I} X_i$ is nonempty.

To me, this settled the question of whether to use Axiom of Choice in practical contexts (although it's still interesting to consider systems of math where it doesn't hold, and the interdependence of various other theorems/results/lemmas/axioms on $AC$).

My first question is:

**Question 1--Is there any similarly fundamental lemma or theorem which depends on the continuum hypothesis or its negation? That is, are there any basic facts in set theory, topology, measure theory, etc. which are (a) "self-evident" and (b) equivalent to $CH$ or $\lnot CH$?**

I would also be interested in hearing if such a statement existed for $GCH$ or its negation $\lnot GCH$, although to me $GCH$ seems "less likely" to be true than $CH$ just because it makes a much broader statement over the class of *all* cardinals, whereas $CH$ is a relatively narrow statement about the relationship of two cardinals $2^{\aleph_0}$ and $\aleph_1$.

Currently, the two "simplest" results (that I know of) in this vein that would directly depend on $CH$ or $\lnot CH$ are:

But neither of these seem intuitively true or false, much less so essential that we had better accept them one way or another if we want to get any serious math done in the related field.

I'm aware that attempts have been made to resolve $CH$ one way or another (e.g. Freiling's axiom of symmetry) that are basically trying to reduce $CH$ to such an obviously true/false statement of general set theory/topology/measure theory. So I have a follow-up:

**Question 2--What seem to be the obstacles to finding such a resolution of $CH$ or $\lnot CH$? That is, why is it so difficult to make concrete and testable statements (i.e. not trivial things like "There exists an element of $2^{2^{\aleph_0}}$ which is neither countable nor of size $\mathfrak{c}$") dependent on $CH$'s truth or falsity? And, should this difficulty be taken as evidence one way or the other for $CH$? Slash, is it actually considered evidence one way or the other for $CH$?**

For instance: every Borel set is either of size $\aleph_0$ (if countable) or of size $2^{\aleph_0}$ (if uncountable). Is our difficulty in constructing a set of intermediate cardinality (as opposed to the ease with which we can construct a non-measurable set) evidence that no such intermediate-cardinality set exists?

I'll also mention that I take the "Platonic view" of $CH$. That is, I believe that despite the existence of models of set theory where either $CH$ or $\lnot CH$ holds, the statement

"If $S = 2^\Bbb{N}$ is the set of all subsets of $\Bbb{N}$, then for $A \subset S$ any subset of $S$, either $A$ is countable, or there exists a 1-1 correspondence between $A$ and $S$"

has a canonical and demonstrable true/false answer.

isa choice function. $\endgroup$3more comments