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

摘要

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 θ~κ < λ for every θ< V): if A is an inverse κ-system of abelian groups having cardinality <λ, then there is a model M such that card(M) = λ and No(M) = card(Gr(A)/Fact(A)). 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 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 <λ such that card(Gr(A)/Fact(A)) = 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(M) = λ is possible when θ~κ < λ for every θ < λ.