This question is about logical complexity of sentences in third order arithmetic. See Wikipedia for the basic concepts.
Recall that the Continuum Hypothesis is a $\Sigma^2_1$ sentence. Furthermore (loosely speaking) it can't be reduced to a $\Pi^2_1$ sentence, as stated in Emil Jeřábek's answer to Can we find CH in the analytical hierarchy?.
Is there an example of a $\Sigma^2_2$ sentence with no known reduction to a $\Pi^2_2$ sentence? (Equivalently, a $\Pi^2_2$ sentence with no known reduction to a $\Sigma^2_2$ sentence.) I mean that there should be no known reduction even under large cardinal assumptions.
I'd prefer an example that's either famous or easy to state. But to begin, any example will do.
Update: Sentences such as "$\mathfrak{c} \leqslant \aleph_2$" and "$\mathfrak{c}$ is a successor cardinal" are $\Delta^2_2$, meaning that they're simultaneously $\Sigma^2_2$ and $\Pi^2_2$. The reason is that each such sentence (and also its negation) can be expressed in the form "$\mathbb{R}$ has a well-ordering $W$ such that $\phi(W)$" where $\phi$ is $\Sigma^2_2$.
Best Answer
The Suslin hypothesis is $\Pi^2_2,$ and $T = ZFC + GCH + LC$ (LC an arbitrary large cardinal axiom) does not prove it to be equivalent to any $\Sigma^2_2$ sentence. Suppose toward contradiction $T$ proves SH to be equivalent to $\exists A \subset \mathbb{R} \varphi(A),$ where $\varphi$ is $\Pi^2_1.$ Assume $V \models T.$
We'll use several results from Chapters VIII and X of Devlin and Johnsbraten's The Souslin Problem. There are generic extensions $V[G] \models T+\diamondsuit^*$ and $V[G][H] \models T+SH$ which do not add reals to or collapse cardinals of $V.$ In $V[G][H],$ there is $A \subset \mathbb{R}$ such that $\varphi(A)$ holds and $A' \subset \omega_1$ which codes a bijection between $\mathbb{R}$ and $\omega_1$ as well as $A.$ By downwards absoluteness of $\varphi,$ $A$ witnesses that $V[G][A'] \models SH.$ But we also have $V[G][A'] \models \diamondsuit^*$ by Lemma 4 (pg. 79), which is a contradiction since $\diamondsuit$ negates SH.