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On power-like models for hyperinaccessible cardinals

Published online by Cambridge University Press:  12 March 2014

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

Extract

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1972

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References

[1]Fuhrken, G., Languages with quantifier “there exist at least ℵα”,The theory of models, North-Holland, Amsterdam, 1965, pp. 121131.Google Scholar
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[8]Schmerl, J. H., On κ-like models for inaccessible κ. Doctoral Dissertation, University of California, Berkeley, 1971.Google Scholar
[9]Schmerl, J. H., An elementary sentence which has ordered models, this Journal, vol. 37 (1972), pp. 521530.Google Scholar
[10]Schmerl, J. H. and Shelah, S., On models with orderings, Notices of the American Mathematical Society, vol. 16 (1969), p. 840. Abstract #69T-E50.Google Scholar
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[12]Shelah, S., On models with power-like orderings, this Journal, vol. 37 (1972), pp. 247267.Google Scholar
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[14]Vaught, R. L., The completeness of logic with the added quantifier “there are uncountably many”, Fundamenta Mathematicae, vol. 54 (1964), pp. 303304.CrossRefGoogle Scholar
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