Sh:644

• Shelah, S., & Väisänen, P. (2000). On inverse \gamma-systems and the number of L_{\infty\lambda}-equivalent, non-isomorphic models for \lambda singular. J. Symbolic Logic, 65(1), 272–284. arXiv: math/9807181 DOI: 10.2307/2586536 MR: 1782119
• Abstract:
  Suppose \lambda is a singular cardinal of uncountable cofinality \kappa. For a model M of cardinality \lambda, let No(M) denote the number of isomorphism types of models N of cardinality \lambda which are L_{\infty\lambda}-equivalent to M. In [Sh:189] inverse \kappa-systems A of abelian groups and their certain kind of quotient limits Gr(A)/Fact(A) were considered. It was proved that for every cardinal \mu there exists an inverse \kappa-system A such that A consists of abelian groups having cardinality at most \mu^\kappa and card(Gr(A)/Fact(A))=\mu. In [Sh:228] a strict connection between inverse \kappa-systems and possible values of No was proved. In this paper we show: for every nonzero \mu\leq\lambda^\kappa there is an inverse \kappa-system A of abelian groups having cardinality <\lambda such that card(Gr(A)/Fact(A))= \mu (under the assumptions 2^\kappa<\lambda and \theta^{<\kappa}<\lambda for all \theta<\lambda when \mu>\lambda), with the obvious new consequence concerning the possible value of No. Specifically, the case No(M)=\lambda is possible when \theta^\kappa<\lambda for every \theta<\lambda.
• Version 1998-07-09_11 (12p) published version (14p)

@article{Sh:644,
 author = {Shelah, Saharon and V{\"a}is{\"a}nen, Pauli},
 title = {{On inverse $\gamma$-systems and the number of $L_{\infty\lambda}$-equivalent, non-isomorphic models for $\lambda$ singular}},
 journal = {J. Symbolic Logic},
 fjournal = {The Journal of Symbolic Logic},
 volume = {65},
 number = {1},
 year = {2000},
 pages = {272--284},
 issn = {0022-4812},
 mrnumber = {1782119},
 mrclass = {03C55 (03C75)},
 doi = {10.2307/2586536},
 note = {\href{https://arxiv.org/abs/math/9807181}{arXiv: math/9807181}},
 arxiv_number = {math/9807181}
}