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A proofless proof of the Barwise compactness theorem1

Published online by Cambridge University Press:  12 March 2014

Mark Howard
Mark Howard*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
*
Odyssey Research Associates, 30/A Harris B. Oates Drive, Ithaca, New York 14850-1313.
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Abstract

We prove a theorem (1.7) about partial orders which can be viewed as a version of the Barwise compactness theorem which does not mention logic. The Barwise compactness theorem is easily equivalent to 1.7 + “Every Henkin set has a model”. We then make the observation that 1.7 gives us the definability of forcing for quantifier-free sentences in the forcing language and use this to give a direct proof of the truth and definability lemmas of forcing.

Type
Survey/expository papers
Information
The Journal of Symbolic Logic , Volume 53 , Issue 2 , June 1988 , pp. 597 - 602
Copyright
Copyright © Association for Symbolic Logic 1988

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Footnotes

1

§§1 and 2 are a revision of the last chapter of Howard [1985].

References

REFERENCES

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Howard, M. [1985], Vaught's conjecture and the closed unbounded filter, Ph.D. thesis, University of California, Berkeley, California.Google Scholar
Keisler, H. J. [1971], Model theory for infinitary logic, North-Holland, Amsterdam.Google Scholar
Shoenfield, J. [1971], Unramified forcing, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, Rhode Island, pp. 357–382.CrossRefGoogle Scholar