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On Shelah’s compactness of cardinals

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An Erratum to this article was published on 01 September 1978


We deal with the compactness property of cardinals presented by Shelah, who proved a compactness theorem for singular cardinals. We improve that result in eliminating axiom I there and show a new application of that theorem together with a straightforward proof of it for the special case discussed. We discuss compactness for regular cardinals and show some independence results: one of them, a part of which is due to A. Litman, is the independence from ZFC+GCH of the gap-one two cardinal problem for singular cardinals.

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This paper is based on the author’s M.Sc. thesis written at The Hebrew University under the supervision of Prof. Shelah, to whom he expresses his deep gratitude.

An erratum to this article is available at

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Ben-David, S. On Shelah’s compactness of cardinals. Israel J. Math. 31, 34–56 (1978).

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