# 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)

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} }