Journal of Symbolic Logic

Diverse Classes

John T. Baldwin

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Abstract

Let $\mathbf{I}(\mu,K)$ denote the number of nonisomorphic models of power $\mu$ and $\mathbf{IE}(\mu,K)$ the number of nonmutually embeddable models. We define in this paper the notion of a diverse class and use it to prove a number of results. The major result is Theorem B: For any diverse class $K$ and $\mu$ greater than the cardinality of the language of $K$, $\mathbf{IE}(\mu,K) \geq \min(2^\mu,\beth_2).$ From it we deduce both an old result of Shelah, Theorem C: If $T$ is countable and $\lambda_0 > \aleph_0$ then for every $\mu > \aleph_0,\mathbf{IE}(\mu,T) \geq \min(2^\mu,\beth_2)$, and an extension of that result to uncountable languages, Theorem D: If $|T| < 2^\omega,\lambda_0 > |T|$, and $|D(T)| = |T|$ then for $\mu > |T|$, $\mathbf{IE}(\mu,T) \geq \min(2^\mu,\beth_2).$

Article information

Source
J. Symbolic Logic, Volume 54, Issue 3 (1989), 875-893.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183743024

Mathematical Reviews number (MathSciNet)
MR1011176

Zentralblatt MATH identifier
0701.03014

JSTOR
links.jstor.org

Citation

Baldwin, John T. Diverse Classes. J. Symbolic Logic 54 (1989), no. 3, 875--893. https://projecteuclid.org/euclid.jsl/1183743024


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