On the number of 𝐿_{∞𝜔₁}-equivalent non-isomorphic models

Shelah, Saharon; Väisänen, Pauli

doi:10.1090/S0002-9947-00-02604-0

On the number of ${L}_{\infty \omega _1}$-equivalent non-isomorphic models
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by Saharon Shelah and Pauli Väisänen

Trans. Amer. Math. Soc. 353 (2001), 1781-1817

DOI: https://doi.org/10.1090/S0002-9947-00-02604-0

Published electronically: December 29, 2000

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Abstract:

We prove that if $\operatorname {ZF}$ is consistent then $\operatorname {ZFC} + \operatorname {GCH}$ is consistent with the following statement: There is for every $k < \omega$ a model of cardinality $\aleph _1$ which is $L_{\infty {\omega _{1}}}$-equivalent to exactly $k$ non-isomorphic models of cardinality $\aleph _1$. In order to get this result we introduce ladder systems and colourings different from the “standard” counterparts, and prove the following purely combinatorial result: For each prime number $p$ and positive integer $m$ it is consistent with $\operatorname {ZFC} + \operatorname {GHC}$ that there is a “good” ladder system having exactly $p^m$ pairwise nonequivalent colourings.

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