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Toward model theory through recursive saturation1

Published online by Cambridge University Press:  12 March 2014

John Stewart Schlipf
John Stewart Schlipf*
Affiliation:
California Institute of Technology, Pasadena CA 91125
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Extract

One of the most significant by-products of the study of admissible sets with urelements is the emphasis it has given to recursively saturated models. As suggested in [Schlipf, 1977], countable recursively saturated models (for finite languages) possess many of the desirable properties of saturated and special models. The notion of resplendency was introduced to isolate some of these desirable properties. In §§1 and 2 of this paper we study these parallels, showing how they can be exploited to give new proofs of some traditional model theoretic theorems. This yields both pedagogical and philosophical advantages: pedagogical since countable recursively saturated models are easier to build and manipulate than saturated and special models; philosophical since it shows that uncountable models — which the downward Lowenheim–Skolem theorem tells us are in some sense not basic in the study of countable theories — are not needed in model theoretic proofs of these theorems. In §3 we apply our local results to get results about resplendent models of ZF set theory and PA (Peano arithmetic). In §4 we shall examine certain analogous results for admissible languages, most similar to, and seemingly generally slightly weaker than, already known results. (The Chang–Makkai sort of result, however, is new.)

Although this paper is an outgrowth of work with admissible sets with urelements, I have tried to keep it as accessible as possible to those with a background only in finitary model theory. Thus §§1,2, and 3 should not involve any work with admissible sets. §4, however, is concerned with some admissible analogues to results in §2 and necessarily uses certain technical results of §11 5 of [Schlipf, 1977].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 2 , June 1978 , pp. 183 - 206
Copyright
Copyright © Association for Symbolic Logic 1978

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Footnotes

1

This paper is a revision of part of the author's Ph.D. thesis at the University of Wisconsin in Madison. The author is indebted to the logicians in Madison for help and inspiration, and particularly to Professor Jon Barwise, his thesis advisor. The preparation of this paper was partially supported by NSF grant MCS 76-17254.

References

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