On inverse $\gamma$-systems and the number of $L_{\infty,\lambda}$-equivalent, non-isomorphic models for $\lambda$ singular

by Shelah and Vaisanen. [ShVs:644]
J Symbolic Logic, 2000
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 <= 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 .

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