TITLE: Class-theoretic sentences that are $\Pi^1_1$ or $\Pi^1_2$
QUESTION [5 upvotes]: I'm looking for the following:
(1) an example of a $\Pi^1_1$ class-theoretic sentence that has no known equivalence to a $\Sigma^1_1$ sentence, even if large cardinal hypotheses or reflection principles are assumed.
(2) an example of a $\Pi^1_2$ class-theoretic sentence that has no known equivalence to a $\Sigma^1_2$ sentence, even if large cardinal hypotheses or reflection principles are assumed.
Admttedly, "large cardinal hypothesis" and "reflection principle" are not well-defined, so these questions are rather vague.
I am assuming impredicative Comprehension (Kelley-Morse class theory) and all forms of choice.
The best answer I currently have for (1) is Vopenka's principle ("every proper class of directed graphs has either a member with a nontrivial endomorphism or two distinct members with a homomorphism between them").  But this is often considered a large cardinal hypothesis, so it's not a good answer.
I don't have any answer for (2).

REPLY [4 votes]: Let's work in GBC for definiteness. In analogy with combinatorics on $\omega_1$, define an $\mathsf{ORD}$-tree to be a (proper class sized) tree of height $\mathsf{ORD}$ so that each level is set sized. Such a tree is Souslin if all of its chains and antichains are set sized as well.
By essentially the same arguments as in the case of any regular $\kappa$, the class-sized forcing consisting of normal binary trees ordered by end-extension adds an $\mathsf{ORD}$-Souslin tree generically while preserving GBC (and KM if it held in the ground model). Moreover the forcing does not add sets so it cannot change any first order property of the model.
Conversely, in unpublished work, Joel David Hamkins and I discovered that any countable model of GBC has an expansion with the same first order part satisfying GBC plus ``there are no $\mathsf{ORD}$-Souslin trees". This model won't satisfy KM even if it held in the ground model, for what it's worth.
The point being, similar to the case of any regular $\kappa$, the existence of an $\mathsf{ORD}$-Souslin tree is independent of GBC and neither the existence of such a tree nor the non-existence of such a tree has any first order consequences over GBC as a base theory.
Now, for a fixed $\mathsf{ORD}$-tree $T$, to say that "$T$ is Souslin" is a $\Pi^1_1$ statement (with $T$ as a parameter of course) since we're really saying ``for all $A \subseteq T$ if $A$ is a chain or an antichain then it is set sized". It's also properly $\Pi^1_1$. If it were equivalent to a $\Sigma^1_1$ sentence then it would be absolute to class-forcing extensions preserving GBC (or KM if that's your ground model theory). But, since we can add a class-sized branch (and antichain) by forcing with $T$, just like in the case on $\omega_1$ this is not true. Some work is needed to show that forcing with the tree preserves GBC/KM but this is in fact true and is essentially a consequence of the fact that the tree is $\mathsf{ORD}$-c.c. and adds no sets.
It follows that to say, for a fixed $\mathsf{ORD}$-tree $T$ that "$T$ is not Souslin" is properly $\Sigma^1_1$.
Clearly we have that to say "there is an $\mathsf{ORD}$-Souslin tree" is $\Sigma^1_2$. I would assume, given the above that it is properly so, but I haven't been able to dot the i's and cross the t's on such a proof.