On inverse γ-systems and the number of L_∞λ-equivalent, non-isomorphic models for λ singular

Abstract

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model of cardinality λ, let No() denote the number of isomorphism types of models of cardinality λ which are L_∞λ-equivalent to . In [7] Shelah considered inverse κ-systems of abelian groups and their certain kind of quotient limits Gr()/ Fact(). In particular Shelah proved in [7, Fact 3.10] that for every cardinal Μ there exists an inverse κ-system such that consists of abelian groups having cardinality at most Μ^κ and card(Gr()/ Fact()) = 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 θ^κ < λ for every θ < λ): if is an inverse κ-system of abelian groups having cardinality < λ, then there is a model such that card() = λ and No() = card(Gr()/ Fact()). The following was an immediate consequence (when θ^κ < λ for every θ < λ): for every nonzero Μ < λ or Μ = λ^κ there is a model , of cardinality λ with No() = Μ. In this paper we show: for every nonzero Μ ≤ λ_κ there is an inverse κ-system of abelian groups having cardinality < λ such that card(Gr()/ Fact()) = Μ (under the assumptions 2^κ < λ and θ^<κ < λ for all θ < λ when Μ > λ), with the obvious new consequence concerning the possible value of No. Specifically, the case No() = λ is possible when θ^κ > λ for every λ < λ.