Journal of Symbolic Logic

Notions of Relative Ubiquity for Invariant Sets of Relational Structures

Paul Bankston and Wim Ruitenburg

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Given a finite lexicon $L$ of relational symbols and equality, one may view the collection of all $L$-structures on the set of natural numbers $\omega$ as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on $\omega$. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on $\omega$ is ubiquitous in the set of linear orderings on $\omega$.

Article information

J. Symbolic Logic, Volume 55, Issue 3 (1990), 948-986.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03C15: Denumerable structures
Secondary: 03C35: Categoricity and completeness of theories 03C52: Properties of classes of models 03C65: Models of other mathematical theories 60B05: Probability measures on topological spaces 90D13 90D45 03C25: Model-theoretic forcing 05C05: Trees 05C40: Connectivity 06A05: Total order 08A55: Partial algebras

Spaces of relational structures ubiquity games Baire category probability complete theories


Bankston, Paul; Ruitenburg, Wim. Notions of Relative Ubiquity for Invariant Sets of Relational Structures. J. Symbolic Logic 55 (1990), no. 3, 948--986.

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