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Löwenheim-Skolem Theorems

  • Kenneth A. Bowen
Chapter
  • 114 Downloads
Part of the Synthese Library book series (SYLI, volume 127)

Abstract

A cardinal constellation is a map c defined on a set X such that for each x ∈ X, c(x) is a non-zero cardinal number. The constellation of a modal structure 𝕬=<A k, K, R, O, N>, c 𝕬, is that constellation c with domain K such that for kK, c(k) = card (|𝒜 k |). Let c and c′ be constellations with domains X and X′, respectively. We write c⊆c′ iff X ⊆ X′ and c′| X = c′, and we write c≤c′ iff X = X′ and for all x ∈ X, c(x) ≤ c′(x). We define card (ML) to be the card inality of the set of all formulas of ML. A cardinal constellation c with domain including K is said to be appropriate for 𝔄 if for all k, k′K, k R k′ implies c(k) ≤ c(k′) and if 𝔄 is a structure for ML, then card(ML) ≤ c(k) for all kK.

Keywords

Modal Structure Free Variable Atomic Formula Individual Constant Elementary Embedding 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1979

Authors and Affiliations

  • Kenneth A. Bowen
    • 1
  1. 1.Syracuse UniversityUSA

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