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Σ1-compactness and ultraproducts

Published online by Cambridge University Press:  12 March 2014

Nigel Cutland
Nigel Cutland*
Affiliation:
University of Hull, Hull, England
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Extract

This paper is devoted to a description of the way in which ultraproducts can be used in proofs of various well-known Σ1-compactness theorems for infinitary languages ℒA associated with admissible sets A; the method generalises the ultra-product proof of compactness for finitary languages.

The compactness theorems we consider are (§2) the Barwise Compactness Theorem for ℒA when A is countable admissible [1], and (§3) the Cofinality (ω) Compactness Theorem of Barwise and Karp [2] and [4]. Our proof of the Barwise theorem unfortunately has the defect that it relies heavily on the Completeness Theorem for ℒA. This defect has, however, been avoided in the case of the Cf(ω) Compactness Theorem, so we have a purely model-theoretic proof.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 4 , December 1972 , pp. 668 - 672
Copyright
Copyright © Association for Symbolic Logic 1972

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References

REFERENCES

[1]Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226–252.Google Scholar
[2]Barwise, J., Strict predicates, this Journal, vol. 34 (1969), pp. 409–423.Google Scholar
[3]Jensen, R. and Karp, C., Primitive recursive set functions, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, Part 1, American Mathematical Society, 1971, pp. 143–176.CrossRefGoogle Scholar
[4]Karp, C., An algebraic proof of the Barwise compactness theorem, The syntax and semantics of infinitary languages, Lecture Notes in Mathematics, vol. 72, Springer-Verlag, Berlin, 1968, pp. 80–95.CrossRefGoogle Scholar
[5]Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.Google Scholar