Journal of Symbolic Logic

On Inverse $\gamma$-Systems and the Number of L$_{\infty\lambda}$- Equivalent, Non-Isomorphic Models for $\lambda$ Singular

Saharon Shelah and Pauli Vaisanen

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

Article information

J. Symbolic Logic, Volume 65, Issue 1 (2000), 272-284.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03C55: Set-theoretic model theory
Secondary: 03C75: Other infinitary logic

Number of Models Infinitary Logic Inverse $\gamma$-System


Shelah, Saharon; Vaisanen, Pauli. On Inverse $\gamma$-Systems and the Number of L$_{\infty\lambda}$- Equivalent, Non-Isomorphic Models for $\lambda$ Singular. J. Symbolic Logic 65 (2000), no. 1, 272--284.

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