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On the number of ${L}_{\infty \omega _1}$-equivalent non-isomorphic models


Authors: Saharon Shelah and Pauli VΓ€isΓ€nen
Journal: Trans. Amer. Math. Soc. 353 (2001), 1781-1817
MSC (2000): Primary 03C55; Secondary 03C75, 03E05
DOI: https://doi.org/10.1090/S0002-9947-00-02604-0
Published electronically: December 29, 2000
MathSciNet review: 1707477
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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 β€œstandard” 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 β€œgood” ladder system having exactly $p^m$ pairwise nonequivalent colourings.


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Additional Information

Saharon Shelah
Affiliation: Institute of Mathematics, The Hebrew University, Jerusalem, Israel and Rutgers University, Hill Ctr-Busch, New Brunswick, New Jersey 08903
MR Author ID: 160185
ORCID: 0000-0003-0462-3152
Email: shelah@math.huji.ac.il

Pauli VΓ€isΓ€nen
Affiliation: Department of Mathematics, P.O. Box 4, 00014 University of Helsinki, Finland
Email: pauli.vaisanen@helsinki.fi

Keywords: Number of models, ladder system, uniformization, infinitary logic, iterated forcing
Received by editor(s): April 28, 1997
Published electronically: December 29, 2000
Additional Notes: The first author thanks GIF for its support of this research, and also the University of Helsinki for funding a visit of the first author to Helsinki in August 1996. This is his paper number 646.
This paper is the second author’s Licentiate’s thesis. The second author did his share of the paper under the supervision of Tapani Hyttinen.
Article copyright: © Copyright 2000 American Mathematical Society