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I’m looking for a survey of monadic second-order theories of the reals.

I’m starting from a 1985 survey by Gurevich which says (p 505) that true arithmetic can be reduced to “the monadic theory of the real line” even though true arithmetic can not be interpreted in that theory. But which monadic theory does he mean? And is there a good reference for the latter claim, for which he cites a 1981 preprint with Shelah?

I think he means the theory in the language with two predicates, one for inclusion and one for openness of subsets. In any case I’d like a reference that covers monadic second-order theories with < and + as well.

Most work on monadic second-order logic seems to focus on orders and trees, which makes it harder to find what I’m looking for. Is there a good reference on monadic second-order theories of the reals from more recently than 1985?

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  • $\begingroup$ This paper by Gurevich and Shelah seems to be relevant. $\endgroup$
    – Levon Haykazyan
    Commented May 31, 2019 at 3:15
  • $\begingroup$ @LevonHaykazyan, thanks, that paper gives the precise statement I was looking for! But it gives the proof with no outline and no other assistance for the reader. $\endgroup$
    – user44143
    Commented May 31, 2019 at 6:44
  • $\begingroup$ Lifsches and Shelah have a more recent and more readable overview for that particular claim in "Peano Arithmetic may not be interpretable in the monadic theory of linear orders", J Symbolic Logic 62 (1997) 848-872, on JSTOR (jstor.org/stable/2275575) or on Shelah's site (shelah.logic.at/short400.html) as paper 471. $\endgroup$
    – user44143
    Commented Jul 9, 2019 at 19:27

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