Even if the answer were negative for arithmetics(I have no idea), in the more general case: Can any mathematical statement be expressed as a $\Delta_m^n$ (with n, m belongs to N) statement in a chosen language? Or are there statements of set theory that cannot be translated into $\Delta_m^n$ statements? (clarification: by $\Delta_m^n$ I mean either $\Sigma_m^n$ or $\Pi_m^n$, not necessarily both)
Sorry I just realized how confusing this question is (or I mean, not what I wanted to ask originally, as for some reason there is a bug in my brain that makes me think that the analytical hierarchy is the same as arbitrary high-order logic), So I will put the questions as answered and restate the question in a form that (I hope) makes sense.
There is no sentence in the analytical hierarchy that is provably equivalent to the generalized continuum hypothesis. ("Provably" means in the usual axiomatization of set theory, ZFC.)