TITLE: For which Millennium Problems does undecidable -> true?
QUESTION [36 upvotes]: Three good answers were received — by Alex Gavrilov, Bjørn Kjos-Hanssen, and Terry Tao — and the bounty has been awarded (somewhat arbitrarily) to Alex Gavrilov.  
The answers are summarized below; because they are open-ended and technically subtle, the question has been flagged for conversion to community Wiki.
Thanks are extended to all who contributed.

Summary  Harry Altman cogently commented:

This is essentially asking which of these statements are equivalent to a $\Pi^0_1$ statement, right? 

We embed this insight into a better version of the question:  

Which of the Millennium Prize problems can be stated as a postulate that can be falsified by a $\Pi^0_1$ counterexample?

to which the answers given (as I understand them) amount to:

"The Riemann Hypothesis is true" …a $\Pi^0_1$ counterexample could exist;(per Noam Elkies' comment)
"The Birch and Swinnerton-Dyer Conjecture is true" … conceivably a $\Pi^0_1$ counterexample could be constructed, but not with present knowledge (per Alex Gavrilov's answer);
"$\mathsf{P}\ne\mathsf{NP}$" … no obvious $\Pi^0_1$ counterexample(per Bjørn Kjos-Hanssen's answer);
"Navier–Stokes is globally regular" … no obvious $\Pi^0_1$ counterexample(per Terry Tao's answer);
"Yang–Mills has a mass gap" … no obvious $\Pi^0_1$ counterexample (?);
"The Hodge Conjecture is true" … no obvious $\Pi^0_1$ counterexample (?);

Resource  Wikipedia's article Arithmetical Hierarchy explains the notation of Harry Altman's answer.
What "No Obvious $\Pi^0_1$ Counterexample" Means   As was noted on Dick Lipton and Ken Regan's weblog Gödel's Lost Letter and P=NP, the authority of the Scientific Advisory Board (SAB) of the Clay Mathematics Institute (CMI) extends to:

"The SAB may consider recommending the award of the prize to an individual who has published work that, in the judgement of the SAB, fully resolves the questions raised by one of the Millennium Prize Problems even if it does not exactly meet the wording in the official problem description.” 

In view of the CMI/SAB's supreme executive authority, the logical possibility of amending a Millennium Prize question to accommodate $\Pi^0_1$ counterexamples — via ingenious "burning arrows," to adopt Dick Lipton and Ken Regan's phrase — cannot be formally excluded.

REPLY [19 votes]: Global regularity for Navier-Stokes on the torus is (logically equivalent to) a $\Pi_2^0$ statement; this is essentially an unpublished observation of Bourgain (who made it in the more general context of supercritical equations), which I sketched out in this paper http://arxiv.org/abs/0710.1604 .  Basically, Navier-Stokes is equivalent to the assertion that for all $T, E > 0$, there exists an $M$ such that all initial data with $H^1$ norm at most $E$, there exists a solution up to time $T$ whose $H^1$ norm is always bounded by $M$.  Now if such a claim is the case, it can be verified (using rigorous perturbation theory) by constructing a sufficient number of approximate solutions to a sufficient number of choices of initial data, and verifying that all of these approximate solutions have norm at most $M/2$ (say) up to time $T$, where "sufficient number" is something explicit that depends on $T, E, M$, and the nature of the approximation can be discretised at an explicit scale that also depends on $T,E,M$.  So Navier-Stokes is equivalent to a statement of the form $\forall T,E \exists M: P(T,E,M)$ where $P(T,E,M)$ is something that can be verified in finite time for each $T,E,M$ (which can be taken to be rational numbers).  (In fact, for each $E$ there is an explicit time $T = T(E)$ for which one needs to verify $P(T,E,M)$, and beyond which global regularity is easily obtained from the energy dissipation, so the claim even simplifies a little to $\forall E \exists M: P( T(E), E, M)$.)
It looks very difficult to me, however, to make Navier-Stokes equivalent to a $\Pi^0_1$ statement.  This would basically amount to being able to describe all possible "blowup scenarios" by a countable set, and to be able to determine whether each such blowup scenario can actually happen in finite time.  While some blowup scenarios (particularly "stable", "approximately self-similar" ones) can be described and verified in such a manner, it's not clear whether all of them can.