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Σ1-compactness in languages stronger than

Published online by Cambridge University Press:  12 March 2014

Nigel Cutland
Nigel Cutland*
Affiliation:
University of Hull, N. Humberside, England
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Extract

One of the main results of Barwise [2] (see also [7, Chapter VIII]) showed that the s – reflection principle for a set A is equivalent to Σ1-compactness of . Here A is any transitive p.r. closed set, and is the infinitary language on A which allows conjunction and disjunction over arbitrary sets Φ Є A, and finite quantification.

In this paper we consider languages , where B is a Δ0 subset of A, which is like but we allow quantifiers ∀x and ∃x where x is any set of variables indexed by an element of B. A treatment similar to that of [2] for establishes a sufficient, and in some cases necessary, condition for to be Σ1-compact. The use of infinitary Skolem functions is intrinsic to the method, so to avoid a separate development of the rudiments of the Skolem language we actually define to have b-ary relation and function symbols for every b Є B.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 3 , September 1978 , pp. 508 - 520
Copyright
Copyright © Association for Symbolic Logic 1978

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References

REFERENCES

[1]Barwise, J., Implicit definability and compactness in infinitary languages, The syntax and semantics of infinitary languages, Lecture Notes in Mathematics, vol. 72 (1968), Springer-Verlag, Berlin and New York, pp. 1–35.CrossRefGoogle Scholar
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[7]Barwise, J., Admissible sets and structures, Springer-Verlag, Berlin and New York, 1975.CrossRefGoogle Scholar