Abstract

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model M of cardinality λ, let No (M) denote the number of isomorphism types of models N of cardinality λ which are L ∞λ - equivalent to M. In [7] Shelah considered inverse κ- systems A of abelian groups and their certain kind of quotient limits Gr(A)/ Fact(A). In particular Shelah proved in [7, Fact 3.10] that for every cardinal μ there exists an inverse κ-system A such that A consists of abelian groups having cardinality at most μ κ and card(Gr(A)/Fact(A)) = μ. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that $\theta^\kappa for every $\theta ): if A is an inverse κ- system of abelian groups having cardinality $\theta^\kappa for every $\theta ): for every nonzero $\mu or μ = λ κ there is a model M μ of cardinality λ with No(M μ ) = μ. In this paper we show: for every nonzero μ ≤ λ κ there is an inverse κ-system A of abelian groups having cardinality $2^\kappa and $\theta^{ for all $\theta when $\mu > \lambda$ ), with the obvious new consequence concerning the possible value of No. Specifically, the case No(M) = λ is possible when $\theta^\kappa for every $\theta