### On the number of $L_{\infty,\omega_1}$-equivalent non-isomorphic models

by Shelah and Vaisanen. [ShVs:646]
Transactions American Math Soc, 2001
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.

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