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On power-like models for hyperinaccessible cardinals

Published online by Cambridge University Press:  12 March 2014

James H. Schmerl
Yale University, New Haven, Connecticut 06520
Saharon Shelah
Hebrew University, Jerusalem, Israel


The main result of this paper is the following transfer theorem: If T is an elementary theory which has a κ-like model where κ is hyperinaccessible of type ω, then T has a λ-like model for each λ > card(T). Helling [3] obtained the same conclusion under the stronger hypothesis that κ is weakly compact. Fuhrken conjectured in [1] that the same conclusion would result if κ were merely inaccessible. (He also showed there the connection with a problem about generalized quantifiers.) Thus, our theorem lies properly between Helling's theorem and Fuhrken's conjecture. In [9] it is shown that this theorem is actually the best possible. This theorem and the other results of this paper were announced by the authors in [10].

In §1 we prove as Theorem 1 a slightly stronger form of the above theorem. This theorem is generalized in §2 to Theorem 2 which concerns theories which permit the omitting of types. The methods used in §§1 and 2 are also applicable to problems regarding Hanf numbers as well as to two-cardinal problems.

Research Article
Copyright © Association for Symbolic Logic 1972

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