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The monadic theory of \(\omega_ 2\). (English) Zbl 0549.03010

A classical result of Büchi states that the monadic (second order) theories of the ordinals \(\omega\) and \(\omega_ 1\) are both decidable. It follows from earlier results of Magidor and Shelah that there are universes of set theory in which the monadic theory of \(\omega_ 2\) is likewise decidable. In the present paper it is shown (assuming the existence of a weakly compact cardinal) that there is an algorithm \(n\mapsto\Psi_ n\) such that \(\Psi_ n\) is a sentence in the monadic language of order and for every \(S\subseteq\omega \) there is a generic extension of the ground world with \(S=\{n;\omega_ 2\vDash\Psi_ n\}\). Thus the monadic theory of \(\omega_ 2\) depends on the underlying set theory and there are continuum many possible monadic theories of \(\omega_ 2\) (in different universes of set theory).
Another result (again proved by forcing) states that each countable model of ZFC\(+''there\) exists a weakly compact cardinal” can be extended to a universe of set theory in which the full second order theory of \(\omega_ 2\) is interpretable (and hence recursive) in the monadic theory of \(\omega_ 2\). Finally it is shown that the monadic theory of \(\omega_ 2\) may have any prescribed complexity in suitable models of set theory.
Reviewer: U.Felgner

MSC:

03B25 Decidability of theories and sets of sentences
03C55 Set-theoretic model theory
03E35 Consistency and independence results
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References:

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