Sh:646

• Shelah, S., & Väisänen, P. (2001). On the number of L_{\infty\omega_1}-equivalent non-isomorphic models. Trans. Amer. Math. Soc., 353(5), 1781–1817. arXiv: math/9908160 DOI: 10.1090/S0002-9947-00-02604-0 MR: 1707477
• Abstract:
  We prove that if ZF is consistent then ZFC + 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 ZFC + GCH that there is a “good” ladder system having exactly p^m pairwise nonequivalent colourings.
• Version 1999-08-10_11 (43p) published version (37p)

@article{Sh:646,
 author = {Shelah, Saharon and V{\"a}is{\"a}nen, Pauli},
 title = {{On the number of $L_{\infty\omega_1}$-equivalent non-isomorphic models}},
 journal = {Trans. Amer. Math. Soc.},
 fjournal = {Transactions of the American Mathematical Society},
 volume = {353},
 number = {5},
 year = {2001},
 pages = {1781--1817},
 issn = {0002-9947},
 mrnumber = {1707477},
 mrclass = {03C75 (03C55 03E05 03E35)},
 doi = {10.1090/S0002-9947-00-02604-0},
 note = {\href{https://arxiv.org/abs/math/9908160}{arXiv: math/9908160}},
 arxiv_number = {math/9908160}
}