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](https://math.stackexchange.com/q/340891) 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.



Can we find CH in the analytical hierarchy?
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.


