Return to Question

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 the negation of each such sentence) is expressible in the form "$\mathbb{R}$ has a well-ordering $W$ such that $\phi(W)$" where $\phi$ is $\Sigma^2_2$.

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 $\Sigma^2_2$.

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 at 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 the negation of each such sentence) is expressible in the form "$\mathbb{R}$ has a well-ordering $W$ such that $\phi(W)$" where $\phi$ is $\Sigma^2_2$.

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 the negation of each such sentence) is expressible in the form "$\mathbb{R}$ has a well-ordering $W$ such that $\phi(W)$" where $\phi$ is $\Sigma^2_2$.

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 at 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 (in outline) is that each of these sentences can be expressed "$\mathbb{R}$ has a well-ordering such that..." and so can its negation.

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 at 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 the negation of each such sentence) is expressible in the form "$\mathbb{R}$ has a well-ordering $W$ such that $\phi(W)$" where $\phi$ is $\Sigma^2_2$.