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

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- by Saharon Shelah and Pauli Vรคisรคnen PDF
- Trans. Amer. Math. Soc.
**353**(2001), 1781-1817 Request permission

## 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 โstandardโ 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 โgoodโ ladder system having exactly $p^m$ pairwise nonequivalent colourings.## References

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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 Vรคisรคnen**- Affiliation: Department of Mathematics, P.O. Box 4, 00014 University of Helsinki, Finland
- Email: pauli.vaisanen@helsinki.fi
- 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 authorโs Licentiateโs thesis. The second author did his share of the paper under the supervision of Tapani Hyttinen. - © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1707477