Return to Question

I'm looking for an open problem in analysis or number theory with just one "genuine" or "second order" quantifier.

E.g.

• "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

• "Every set $S$ of natural numbers has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

No cheat examples like "For every real number, Goldbach's conjecture holds"! That's an arithmetical problem.

In technical terms, I'm looking for a $\Pi^1_1$ sentence that we don't know how to reduce to an arithmetical sentence.

CLARIFICATION (added 16 August 2021)

I have examples of such sentences now, but they require specialist background to understand (e.g. functional analysis).

What I'm really looking for is an example that is easy to state and obviously $\Pi^1_1$ for readers without specialist background. (And has no known reduction to an arithmetical sentence.)

While I'd prefer a known problem, I'll settle for a contrived sentence that no mathematician would care about.

I'm looking for an open problem in analysis or number theory with just one "genuine" or "second order" quantifier.

E.g.

• "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

• "Every set $S$ of natural numbers has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

No cheat examples like "For every real number, Goldbach's conjecture holds"! That's an arithmetical problem.

In technical terms, I'm looking for a $\Pi^1_1$ sentence that we don't know how to reduce to an arithmetical sentence.

I'd also like it to be easy to state and obviously $\Pi^1_1$, so that it can be included in a logic paper without requiring much explanation.

I'm looking for an open problem in analysis or number theory with just one "genuine" or "second order" quantifier.

E.g.

• "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.S$

• "Every set $S$ of natural numbers has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

No cheat examples like "For every real number, Goldbach's conjecture holds"! That's an arithmetical problem.

In technical terms, I'm looking for a $\Pi^1_1$ sentence that we don't know how to reduce to an arithmetical sentence.

CLARIFICATION (added 16 August 2021)

I have examples of such sentences now, but they require specialist background to understand (e.g. functional analysis).

What I'm really looking for is an example that is easy to state and obviously $\Pi^1_1$ for readers without specialist background. (And has no known reduction to an arithmetical sentence.)

While I'd prefer a known problem, I'll settle for a contrived sentence that no mathematician would care about.

I'm looking for an open problem in analysis or number theory with just one "genuine" or "second order" quantifier.

E.g.

• "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

• "Every set $S$ of natural numbers has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

No cheat examples like "For every real number, Goldbach's conjecture holds"! That's an arithmetical problem.

In technical terms, I'm looking for a $\Pi^1_1$ sentence that we don't know how to reduce to an arithmetical sentence.

CLARIFICATION (added 16 August 2021)

I have examples of such sentences now, but they require specialist background to understand (e.g. functional analysis).

What I'm really looking for is an example that is easy to state and obviously $\Pi^1_1$ for readers without specialist background. (And has no known reduction to an arithmetical sentence.)

While I'd prefer a known problem, I'll settle for a contrived sentence that no mathematician would care about.

I'm looking for an open problem in analysis with just one "genuine" quantifier.

E.g.

• "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

No cheat examples like "For every real number, Goldbach's conjecture holds"! That's a problem in number theory, not a problem in analysis.

In technical terms, I'm looking for a $\Pi^1_1$ sentence that we don't know how to reduce to an arithmetical sentence.

CLARIFICATION (added 16 August 2021)

I have an examples of such sentences now, but they require specialist background to understand (e.g. functional analysis).

What I'm really looking for is an example that is easy to state and obviously $\Pi^1_1$ for readers without specialist background. (And has no known reduction to an arithmetical sentence.)

While I'd prefer a known problem, I'll settle for a contrived sentence that no mathematician would care about.

I'm looking for an open problem in analysis or number theory with just one "genuine" or "second order" quantifier.

E.g.

• "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.S$

• "Every set $S$ of natural numbers has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

No cheat examples like "For every real number, Goldbach's conjecture holds"! That's an arithmetical problem.

In technical terms, I'm looking for a $\Pi^1_1$ sentence that we don't know how to reduce to an arithmetical sentence.

CLARIFICATION (added 16 August 2021)

I have examples of such sentences now, but they require specialist background to understand (e.g. functional analysis).

What I'm really looking for is an example that is easy to state and obviously $\Pi^1_1$ for readers without specialist background. (And has no known reduction to an arithmetical sentence.)

While I'd prefer a known problem, I'll settle for a contrived sentence that no mathematician would care about.