Invariant Subspace Conjecture: A bounded operator on a separable Hilbert space has a non-trivial closed invariant subspace.
Can this conjecture be reformulated as an arithmetic statement, that is, $\Pi^0_n$ statement for some n? (I tried to figure it out, but failed.)
EDIT: For what I understand from answers, it appears to be an open problem. As Emil Jerabek and others mentioned, the intrinsic complexity of the conjecture (considered as a statement in second-order arithmetic) is $\Pi^1_2$. Apparently, no reduction to lesser complexity is known. One may speculate about how much of a solution would be a reduction to $\Pi^1_1$ or $\Pi^0_n$, but I would rather not.
Carl Mummert pointed out an interesting possibility: whether the conjecture itself is true or not, its interpretation in computable analysis may be false. In this case, if I got it right, the only way to reduce its complexity is to disprove it. However, this obstacle would disappear if we are allowed to use set theory to prove equivalence, because computable analysis doesn't work there.
Thanks to everyone.