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Σ1-compactness and ultraproducts

  • Nigel Cutland (a1)

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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 A 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 A 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 A. This defect has, however, been avoided in the case of the Cf(ω) Compactness Theorem, so we have a purely model-theoretic proof.

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[1]Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226252.
[2]Barwise, J., Strict predicates, this Journal, vol. 34 (1969), pp. 409423.
[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. 143176.
[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. 8095.
[5]Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.

Σ1-compactness and ultraproducts

  • Nigel Cutland (a1)

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