Abstract
The main goal of this paper is to complete the classification of those first-order theories such that\(I_{N_0 }^a (\lambda ,T) = 2^\lambda \). We introduce two notions, thedimensional order property anddeepness. Our main theorem asserts that for a superstable theory\(T,I_{N_{\text{0}} }^a (\lambda ,T) = 2^\lambda \) iffT has the dimensional order property or is deep. In a sense made precise in §4 this provides a syntactical characterization of theories with the maximum number of ℵɛ-saturated models in each power.
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The author would like to thank John Baldwin for patiently reading the handwritten manuscript and helping to improve the presentation, and to Baldwin and Makkai for many more remarks later.
This research was supported by the United States—Israel Binational Science Foundation.
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Shelah, S. The spectrum problem I: ℵɛ-satuarted models, the main gap. Israel J. Math. 43, 324–356 (1982). https://doi.org/10.1007/BF02761237
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DOI: https://doi.org/10.1007/BF02761237