TITLE: Can we find CH in the analytical hierarchy?
QUESTION [19 upvotes]: Recently I was talking to my friend and I have mentioned to him that it was proven that CH is not provably (over ZFC) equivalent to any statement in second-order arithmetic. However, today I found out that the result I was thinking about is the one mentioned in this answer, which says that GCH is independent of all sentences in analytical hierarchy. This now made me think:

Is there a sentence in the analytical hierarchy which is provably equivalent to continuum hypothesis?

I am expecting the answer to this question to be no, because CH concerns arbitrary sets of reals, but this is only an intuitive reason.
Thanks in advance.

REPLY [24 votes]: The collapsing forcing by countable partial functions from $\omega_1$ to $2^\omega$ is $\omega_1$-closed, hence it preserves $H_{\omega_1}$, and a fortiori the truth of all formulas in the analytical hierarchy; it also makes CH hold. Thus CH is not equivalent to any statement in the analytical hierarchy (assuming ZFC is consistent).
In fact, the argument implies that ZFC+CH is $\Pi^2_1$-conservative over ZFC, hence CH is not equivalent to a $\Pi^2_1$ statement. On the other hand, it is easy to see that it can be formulated as a $\Sigma^2_1$ statement.