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Requirement systems

Published online by Cambridge University Press:  12 March 2014

Julia F. Knight
Julia F. Knight*
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
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Methods for carrying out transfinitely nested priority constructions have been developed by Harrington [7] and by Ash [2, 1, 3, 4]. Ash's method has different versions, with later ones becoming simpler. Lemmp and Lerman [11] have also developed a method, for finitely nested constructions. Ash formulated abstractly the object of a nested priority construction, and he proved a metatheorem for what he called “α-systems”, listing conditions which guarantee the success of the construction. Harrington's method of “workers”, at least in its original, informal state, seems more flexible than Ash's α-systems.

In [10, 9], there are finite and transfinite versions of a metatheorem for workers. The statements are complicated, and these metatheorems have not proved to be very useful. The present paper gives a new transfinite metatheorem. The statement is considerably simpler than the one in [9], although not so simple as that in [3]. The new metatheorem grew, in part, out of an effort to find a new proof of Ash's metatheorem. The new metatheorem yields the one in [3], and it seems more flexible. A different generalization of Ash's metatheorem will be given in [5].

Ash's metatheorem is easier to use than the one in the present paper, and the result in [3] is certainly the one to use wherever it applies. Here we give one application of the new metatheorem which does not seem to follow from the result in [3]. This is a theorem on models “representing” a given Scott set, which implies one half of a recent result of Solovay [18], on Turing degrees of models of particular completions of Peano arithmetic (PA).

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Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 1 , March 1995, pp. 222 - 245
Copyright
Copyright © Association for Symbolic Logic 1995

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References

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