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The complete extensions of the monadic second order theory of countable ordinals. (English) Zbl 0541.03003
Let MT[co] be the monadic second order theory of countable ordinals. The authors characterize the complete extensions of MT[co] by axiom systems. They introduce the principle of definable choice, and show that the same elements are definable in the monadic and in the elementary case. They distinguish three types of extensions according to the order type of the definable elements. The authors show that only the extensions in \({\mathfrak Q}_ 0\) satisfy the principle of definable choice. They generalize the notion of ultimately periodic sets to the nonstandard ordinals which together with the ultimately periodic subsets form the prime models of the extensions in \({\mathfrak Q}_ 0\) and \({\mathfrak Q}_ 1\) but no model at all for the extensions in \({\mathfrak Q}_ 2\) (\({\mathfrak Q}\) is a set of functions \(c:\omega +1\to \omega \cup \{\omega +\omega^*\}\) and \({\mathfrak Q}={\mathfrak Q}_ 0\cup {\mathfrak Q}_ 1\cup {\mathfrak Q}_ 2).\) (This paper was written in 1972-1974.)
Reviewer: M.Tetruashvili
MSC:
03B25 Decidability of theories and sets of sentences
03C85 Second- and higher-order model theory
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