I do not understand how the continuum hypothesis can be undecidable

Question

I have been thinking about what it means for the continuum hypothesis to be undecidable.

In short, here is my understanding

$\aleph_{0}$ is the cardinality of the set of integers.

$\aleph_{1}$ is the cardinality of the smallest set bigger than $\aleph_{0}$

According to the continuum hypothesis, $2^{\aleph_{0}}=\aleph_{1}$

Now, here is why I do not understand:

Let us suppose there exists some set S which is greater than $\aleph_{0}$ but smaller than $2^{\aleph_{0}}$

Then, it will be easy to prove by Cantor's diagonal argument that S violates the hypotheses. So, if the continuum hypothesis is false, it must be provably false.

This is imposible if we assume that the continuum hypothesis is undecidable, as it means that the truth of the continuum hypothesis is consistent with ZFC, and there by it cannot be provably false.

Therefore, since if it was false it would be PROVABLY false, which is impossible, the continuum hypothesis is true.

Is there a problem with this reasoning? Have I tripped over my own argument, or is there no reason to assume it is provably false? Please explain.

Answer 1

Your mistake, of course, is that the falsehood of CH need not imply that it is provably false.

Yes, if CH is false, there is some $S\subseteq\Bbb R$ such that $\aleph_0<|S|=\aleph_1<2^{\aleph_0}$. But in different models of ZFC this $S$ might be very different. It is not necessarily that the same definition always works as a counterexample to CH.

In fact, in some models it could be that no definable set of reals is a counterexample to CH, but CH fails nonetheless.