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}
}