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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Degrees of recursively saturated models

Authors: Angus Macintyre and David Marker
Journal: Trans. Amer. Math. Soc. 282 (1984), 539-554
MSC: Primary 03D45; Secondary 03C50, 03C57, 03C62, 03H15
MathSciNet review: 732105
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Abstract: Using relativizations of results of Goncharov and Peretyat'kin on decidable homogeneous models, we prove that if $ M$ is $ S$-saturated for some Scott set $ S$, and $ F$ is an enumeration of $ S$, then $ M$ has a presentation recursive in $ F$. Applying this result we are able to classify degrees coding (i) the reducts of models of PA to addition or multiplication, (ii) internally finite initial segments and (iii) nonstandard residue fields. We also use our results to simplify Solovay's characterization of degrees coding nonstandard models of Th(N).

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Additional Information

Keywords: Scott set, Turing degree, recursively saturated model, model of arithmetic
Article copyright: © Copyright 1984 American Mathematical Society