Approaches to Riemann hypothesis using methods outside number theory

I have no idea to what extent the idea of Saharon Shelah, about which I read in David Ruelle's popular account the mathematician's brain that uses mathematical logic to prove the RH is promising, but certainly it is different. For as far as I can understand (from Ruelle), it basically comes down to proving that RH is undecidable in Peano arithmetic, in which case the consistency of Peano arithmetic would imply its truth (also in ZFC).

EDIT: Here is the quote from Shelah's paper:

2.3 Dream: Prove that the Riemann Hypothesis is unprovable in PA, but is
provable in some higher theory.

What basis does my hope for this dream have? First, the solution of Hilbert’s
10th problem tells us that each problem of the form “is the theory ZFC +φ consistent” can be translated to a (specific) Diophantine equation being unsolvable
in the integers, moreover the translation is uniform (this works for any reasonable
(defined) theory, where consistent means that no contradiction can be proved from
it). Second, we may look at parallel development “higher up”; as the world is quite
ordered and reasonable.

Note that there is a significant difference between $Pi_2$ sentences (which say, e.g.,
for a given polynomial $f$, the sentence $varphi_f$ saying that for all natural numbers
$x_0 , ldots , x_{n−1}$ there are natural numbers $y_0 , ldots , y_m$ such that $f (x_0 , ldots , y_0 , ldots ) = 0$) and $Pi_1$ sentences saying just that, e.g., a certain Diophantine equation is unsolvable. The first ones can be proved not to follow from PA by restricting ourselves to a proper initial “segment” of a nonstandard model of PA. For $Pi_1$ sentences, in some sense proving their consistency show they are true (as otherwise PA is inconsistent). Naturally, concerning statements in set theory, models of ZFC are more malleable, as the method of forcing shows.