Decidability of the cardinality of a set given that the Continuum Hypothesis is independent from ZFC

Question

I'm a Total Amateur (TM), please forgive me if this question makes no sense.

The Continuum Hypothesis states that there are no sets with cardinality strictly between that of the integers and the reals. It's negation says there is some set with such a property. Both possible axioms refer to the same sets, those existing under ZFC. Isn't therefore a simple corollary of the independence of the CH that there must exist sets in ZFC with an undecidable cardinality?
I don't know if this sets can be described without an infinite amount of information, but they must exist. I'm guessing this is pretty trivial, but I've wondered about it. It would be interesting if there existed a set which has a finite description but with a cardinality that cannot be decided. Is there such a set? (I'm not sure if I'm using the concept "decidable" correctly, I mean something like "proving that such and such set has ANY one cardinality is impossible in ZFC") Thanks.

Last minute edit: Wait, I just thought of something: isn't "the set of all sets with a cardinality strictly between that of the integers and the reals" such a set? I mean, you can't determine it's cardinality just with ZFC, it could be empty, or not, depending on adding CH or not! But is this set well defined? And what does "well defined" even mean in the context of a set whose cardinality depends on an undecidable statement? Anyway, thanks for any clarification.

Answer 1

I think the most important concept here is knowing what independence really means. If a theory (like ZFC) proves a statement, then every model of that theory must 'think' that statement is true. If a statement is independent from a theory, then there exists a model of the theory which thinks that statement is true, and there also exists a model of the theory which thinks that statement is false.

A simple example: consider the theory of linear orders. We have a language with one relation, and the theory asserts that the one relation is transitive, reflexive, antisymmetric, and total. Note that the statement "there is a smallest element" is independent of this theory: both the natural numbers and the integers with the usual 'less than' relation are models of this theory, but the natural numbers have a smallest element, while the integers do not. Yet both models contain zero (or one..depending on how you define $\mathbb{N}$)!

When you say "there must exist sets in ZFC with an undecidable cardinality," you are missing the mark. Formulas define sets that exist in models of ZFC. Different models of ZFC can interpret such set-defining formulas differently. Your idea of a "set of all sets with a cardinality strictly between that of the integers and the reals" is a great example. You are right that in a model of ZFC that also thinks CH, this formula would represent the empty set, while in another model of ZFC that does not think CH, this formula would represent a non-empty set.

Answer 2

The set in your edit is just one example of a general phenomenon: for an arbitrary sentence $\varphi$, let $X$ be the set of all $y$ such that

• EITHER $y=0$ and $\varphi$ holds,

• OR $y\not=y$ and $\varphi$ fails.

Then whether $X$ has cardinality 0 or 1 is completely determined by whether $\varphi$ is true or not; and of course there's nothing special about "0" or "1", here. And we can't escape this: by Godel's incompleteness theorem, for any "reasonable" set theory $T$ (for instance, ZFC+GCH) there is a $\varphi$ which is not provable or disprovable in $T$.

Answer 3

ZFC does not precisely specify the universe of sets, just describes some of its perceived properties.

Just like when you ask "is the set of numbers complete" the answer may be different depending on what you mean by "numbers". The reals are complete but the rationals are not.

The universe of sets has never been specified so precisely as to be able to deduce either CH or not CH. Different interpretations result in a different answer.

There are some interpretations (or rather models) of ZFC in which CH holds. In other models it doesn't. But you can't have a model of both, simultaneously. It is not the case that CH and its negation could simultaneously hold, or that you have sets of "undecidable cardinality", in the same model.

An analogy with Euclid's Fifth Postulate is often used. There are geometries in which the fifth postulate holds, and other geometries in which it does not. There is no single geometry in which the fifth postulate both holds and fails (well, unless the axioms imply a contradiction, which we hope is not the case). We do not know if the world has Euclidean or hyperbolic geometry (if this question makes sense at all), and we do not know if the ZFC universe satisfies CH or not (again, if this question makes sense at all, as sets are after all not physical objects).