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An apparent example is "The continuum is an $\aleph$-fixed point," that is $\mathfrak{c}=\aleph_\mathfrak{c}$. This is equivalent of course to $\mathfrak{c}\ge\aleph_\mathfrak{c}$, which means it can be expressed as "There is an $\mathbb{R}$-indexed family of sets of reals of pairwise distinct cardinalities," which is $\Sigma^2_2$. I do not see a $\Pi^2_2$ equivalent.

(A bit more explicitly: "There is a family $(A_r)_{r\in\mathbb{R}}$ of sets of reals such that there is no pair of injections $f:A_r\rightarrow A_s$, $g:A_s\rightarrow A_r$ with $r\not=s$." Via a bijection $\mathbb{R}^2\rightarrow\mathbb{R}$ we can identify an $\mathbb{R}$-indexed family of sets of reals with a single set of reals.)

One level up (I originally miscounted - thanks Andreas) we seem to have "The continuum is a limit cardinal." This can be expressed in a $\Pi^2_3$ way as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_3$ equivalent even granting large cardinals. (In particular, note that "The continuum is greater than or equal to some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)

However, I don't actually see a proof that any of the complexity calculations above are optimal. That said, I think this is a good indication that general continuum combinatorics is a good place to look for high-complexity third-order sentences.

Here are two non-examples, one erring in each direction:

• Too simple: "The continuum is an $\aleph$-fixed point," that is $\mathfrak{c}=\aleph_\mathfrak{c}$. This is equivalent of course to $\mathfrak{c}\ge\aleph_\mathfrak{c}$, which means it can be expressed as "There is an $\mathbb{R}$-indexed family of sets of reals of pairwise distinct cardinalities," which is $\Sigma^2_2$. Contra my original guess, however, this does have a $\Pi^2_2$ equivalent observed by Farmer S in the comments below (and I'll add his argument here later when I have more time).

• Too complicated (so far!): "The continuum is a limit cardinal." This can be expressed in a $\Pi^2_3$ way (which I originally miscounted - thanks to Andreas Blass for bringing this to my attention) as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_3$ equivalent even granting large cardinals. (In particular, note that "The continuum is $\ge$ some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)

Of course, the first example doesn't work, and the second example almost certainly doesn't work. That said, I think this is a good indication that general continuum combinatorics is a good place to look for high-complexity third-order sentences.

An apparent example is "The continuum is an $\aleph$-fixed point," that is $\mathfrak{c}=\aleph_\mathfrak{c}$. This is equivalent of course to $\mathfrak{c}\ge\aleph_\mathfrak{c}$, which means it can be expressed as "There is an $\mathbb{R}$-indexed family of sets of reals of pairwise distinct cardinalities," which is $\Sigma^2_2$. I do not see a $\Pi^2_2$ equivalent.

One level up (I originally miscounted - thanks Andreas) we seem to have "The continuum is a limit cardinal." This can be expressed in a $\Pi^2_3$ way as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_3$ equivalent even granting large cardinals. (In particular, note that "The continuum is greater than or equal to some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)

However, I don't actually see a proof that any of the complexity calculations above are optimal. That said, I think this is a good indication that general continuum combinatorics is a good place to look for high-complexity third-order sentences.

An apparent example is "The continuum is an $\aleph$-fixed point," that is $\mathfrak{c}=\aleph_\mathfrak{c}$. This is equivalent of course to $\mathfrak{c}\ge\aleph_\mathfrak{c}$, which means it can be expressed as "There is an $\mathbb{R}$-indexed family of sets of reals of pairwise distinct cardinalities," which is $\Sigma^2_2$. I do not see a $\Pi^2_2$ equivalent.

(A bit more explicitly: "There is a family $(A_r)_{r\in\mathbb{R}}$ of sets of reals such that there is no pair of injections $f:A_r\rightarrow A_s$, $g:A_s\rightarrow A_r$ with $r\not=s$." Via a bijection $\mathbb{R}^2\rightarrow\mathbb{R}$ we can identify an $\mathbb{R}$-indexed family of sets of reals with a single set of reals.)

One level up (I originally miscounted - thanks Andreas) we seem to have "The continuum is a limit cardinal." This can be expressed in a $\Pi^2_3$ way as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_3$ equivalent even granting large cardinals. (In particular, note that "The continuum is greater than or equal to some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)

However, I don't actually see a proof that any of the complexity calculations above are optimal. That said, I think this is a good indication that general continuum combinatorics is a good place to look for high-complexity third-order sentences.

I believe that "The continuum is a limit cardinal" does the job. This can be expressed in a $\Pi^2_2$ way as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_2$ equivalent even granting large cardinals. (In particular, note that "The continuum is greater than or equal to some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)

Another example is "The continuum is an $\aleph$-fixed point," that is $\mathfrak{c}=\aleph_\mathfrak{c}$. This is equivalent of course to $\mathfrak{c}\ge\aleph_\mathfrak{c}$, which means it can be expressed as "There is an $\mathbb{R}$-indexed family of sets of reals of pairwise distinct cardinalities," which is naively $\Sigma^2_2$.

However, I don't actually see a proof that any of the complexity calculations above are optimal. That said, I think this is a good indication that general continuum combinatorics is a good place to look for high-complexity third-order sentences.

An apparent example is "The continuum is an $\aleph$-fixed point," that is $\mathfrak{c}=\aleph_\mathfrak{c}$. This is equivalent of course to $\mathfrak{c}\ge\aleph_\mathfrak{c}$, which means it can be expressed as "There is an $\mathbb{R}$-indexed family of sets of reals of pairwise distinct cardinalities," which is $\Sigma^2_2$. I do not see a $\Pi^2_2$ equivalent.

One level up (I originally miscounted - thanks Andreas) we seem to have "The continuum is a limit cardinal." This can be expressed in a $\Pi^2_3$ way as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_3$ equivalent even granting large cardinals. (In particular, note that "The continuum is greater than or equal to some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)

However, I don't actually see a proof that any of the complexity calculations above are optimal. That said, I think this is a good indication that general continuum combinatorics is a good place to look for high-complexity third-order sentences.