# Open problem in analysis with just one quantifier?

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.

• Problems of the form: is there a countable graph with such-and-such first-order property? Is there a countable structure with such-and-such first-order property? Negative answers would have the desired form $\Pi^1_1$. Aug 8, 2021 at 12:18
• $\forall (x,y).\zeta(x+iy)=0\implies x={1\over 2}$ is a natural thing to write, but RH turns out to have a $\Pi^1_0$ form. So your question has subtleties.
– none
Aug 8, 2021 at 17:02
• I suspect the four exponentials conjecture and some other problems in transcendental number theory could qualify. Aug 9, 2021 at 15:35
• Sorry, what is meant by "genuine" quantifier? Doesn't the term "continuous function" implicitly contain quite a few quantifiers? (EDIT: Oh I see -- the condition that a given function be continuous is arithmetical, so the only analytic quantifier is over the function itself) Aug 9, 2021 at 15:52
• @TimCampion a continuous function can be presented as a continuous function on rationals. Aug 11, 2021 at 10:11

It is easy to state and obviously $$\Pi^1_1$$. Furthermore this comment by Christian Reiher gives me confidence that it has no known reduction to an arithmetical sentence. (Hopefully it even lacks a known reduction to a $$\Sigma^1_1$$ sentence.)
• Thanks to everyone that helped, either by suggesting possible answers, or by pointing out that various $\Pi^1_1$ sentences suggested by me or others did in fact have a known reduction to an arithmetical sentence. Aug 17, 2021 at 20:39