# 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.
• 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.
• published version (37p)
Bib entry
@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}
}