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This is a follow up to Will Brian's answer to this recent question. In particular, quoting Brian:

"In fact, Paul Larson has pointed out to me that the statement "$\phi$ and $\phi^{-1}$ are conjugate" is a now very rare example of a $\Sigma^2_1$ statement about the real line whose status we do not know under ZFC+CH."

(A $\Sigma^2_1$ statement is one that can be expressed in the form $\exists A\subseteq\mathbb{R}\psi(A,r)$ where $r$ is a fixed real parameter and $\psi$ is a formula whose quantifiers range only over $\mathbb{R}$.)

How rare is "very rare"? Are there any other known examples of $\Sigma^2_1$ statements concerning the real line whose status under CH (+ large cardinals) is unknown? What are some other examples?

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    $\begingroup$ This should say ZFC+CH+large cardinals, or we get silly projective statements. $\endgroup$
    – Andrés E. Caicedo
    Commented Feb 22, 2020 at 2:36
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    $\begingroup$ Well another example, though of the same kind, is the following: let $(p_n)$ and $(p'_n)$ be the increasing sequence of primes $=1$ mod $4$, resp. $=3$ mod $4$ (this precise choice doesn't matter), let $c$ be a permutation with one $p_n$-cycle for each $n$ and no other cycle, and similarly define $c'$. Are $c$ and $c'$ conjugate as self-homeomorphisms of the Stone-Cech remainder? I just guess it's unknown, I'm not sure anyone else ever thought about it. $\endgroup$
    – YCor
    Commented Feb 22, 2020 at 3:33
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    $\begingroup$ As an outsider, I don't guess what's the meaning of "under CH+ large cardinals". Does it mean "under CH, assuming the consistency of existence of some kind (what kind?) of large cardinal"? or *"under CH + existence of some kind (what kind??) of large cardinal"? is the [consistency of the?] existence of a strongly inaccessible cardinal enough to avoid the pathologies mentioned by Andrés? By the way I'd also like to see some "typical" illustrating examples of theorems of ZFC or ZFC+CH reducing to a $\Sigma_1^2$ statement. $\endgroup$
    – YCor
    Commented Feb 22, 2020 at 18:34
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    $\begingroup$ This is all motivated by a result of Woodin that $\Sigma^2_1$ statements are absolute for set forcing between models of ZFC+CH+"enough large cardinals". So if there are enough large cardinals around, there is a kind of "canonical" theory of ZFC+CH at the level of $\Sigma^2_1$. The question of whether $\phi$ and $\phi^{-1}$ are conjugate has the right complexity, but it is unknown what the answer is in the canonical theory. (Also, Andres is much more of an expert on this than I am, so he may be able to say more intelligent things here.) $\endgroup$
    – Todd Eisworth
    Commented Feb 22, 2020 at 18:48
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    $\begingroup$ @AndrésE.Caicedo: Thanks for pointing that out. As you could probably guess, the error is mine, not Paul's. Todd, one answer to your question, which I also got from Paul Larson, is "there is a self-homeomorphism $\psi$ of $\omega^*$ that commutes with the shift map $\phi$ but is not a power of it." Like the example in YCor's comment, I'm not sure this should really qualify as an answer, because it seems a bit too close to the statement we started with. It would be nice to see an example that doesn't just ask about pathological automorphisms of $\mathcal P(\omega)/\mathrm{fin}$. $\endgroup$
    – Will Brian
    Commented Feb 25, 2020 at 13:41

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