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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 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).

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).

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 all forms of choice.

The best answer I currently have for (1) is Vopenka's principle ("every proper class of directed graphs has 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).

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 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).

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 all forms of choice.

The best answer I currently have for (1) is Vopenka's principle ("every proper class of directed graphs has 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 example of (2).

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 all forms of choice.

The best answer I currently have for (1) is Vopenka's principle ("every proper class of directed graphs has 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).