Gurevich and Shelah showed in The monadic theory and the “next world” that the monadic theory of the real line (or even just the Cantor Discontinuum) can compute
- the $V^B$ theory of second order arithmetic,
- the $V^B$ theory of third order arithmetic if CH holds (or if the union of $<\!c$ meager sets is meager),
where $B$ is the Boolean algebra of regular open sets, equivalently the algebra for adding a Cohen real.
Note the sharp contrast with the decidability of S2S.
Also, despite its power, the monadic theory of the real line does not interpret (allowing parameters) even Robinson arithmetic $\mathrm{Q}$ (On the strength of the interpretation method by Gurevich and Shelah; the theory there is mutually interpretable with $\mathrm{Q}$, $\mathrm{I}Σ_0$, and other typical weak arithmetical theories without exponentiation). See also Peano Arithmetic may not be interpretable in the monadic theory of orders by Lifsches and Shelah. The reason is that we have certain symmetries that break any purported interpretation.
However, the Cohen algebra $B$ is interpretable in the monadic theory of the reals, and with some effort, the theory can be used to reason about $B$-valued models, hence the unusual form of the complexity result.
I suspect the qualification $V^B$ can be removed in the 'true' $V$, but I only know the trivial (given known absoluteness results) consequences:
- under $V=L$, the theory of third order arithmetic is computable from the theory,
- under projective determinacy, the theory of second order arithmetic is computable from the theory,
- under a measurable Woodin cardinal + CH, $Σ^2_1$ truth is computable from the theory,
- there is a generic extension of $V$ (such as $\mathrm{Add}(ω_1,1) \times \mathrm{Add}(ω,1)$) in which the theory of third order arithmetic is computable from the theory (both as defined in the extension).
