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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A two-cardinal theorem


Author: Saharon Shelah
Journal: Proc. Amer. Math. Soc. 48 (1975), 207-213
DOI: https://doi.org/10.1090/S0002-9939-1975-0357105-9
MathSciNet review: 0357105
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Abstract: We prove the following theorem and deal with some related questions: If for all $n < \omega ,T$ has a model $M$ such that ${n^n} \leq |{Q^M}{|^n} \leq |{P^M}| < {\aleph _0}$ then for all $\lambda ,\mu$ such that $|T| \leq \mu \leq \lambda < {\operatorname {Ded} ^ \ast }(\mu )$ (e.g. $\mu = {\aleph _0},\lambda = {2^{{\aleph _0}}}), T$ has a model of type $(\lambda ,\mu )$, i.e. $|{Q^M}| = \mu ,|{P^M}| = \lambda$.


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Keywords: Two-cardinal theorem, finite models
Article copyright: © Copyright 1975 American Mathematical Society