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Comparing Notions of Similarity For Uncountable Models

Taneli Huuskonen
The Journal of Symbolic Logic
Vol. 60, No. 4 (Dec., 1995), pp. 1153-1167
DOI: 10.2307/2275879
Stable URL: http://www.jstor.org/stable/2275879
Page Count: 15
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Comparing Notions of Similarity For Uncountable Models
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Abstract

The present article, which is a revised version of part of [Hu1], deals with various relations between models which might serve as exact formulations for the vague concept "similar" or "almost isomorphic". One natural class of such formulations is equivalence in a given logic. Another way to express similarity is by potential isomorphism, i.e., isomorphism in some extension of the set-theoretic universe. The class of extensions may be restricted to give different notions of potential isomorphism. A third method is to study the winning strategies for an Ehrenfeucht-Fraisse-game played between the two models, and the properties of the resulting equivalence and nonequivalence trees. The basic question studied here is whether one such notion of similarity implies another. Some implications and counterexamples listed in this part are previously known or trivial, but all are mentioned for completeness' sake. Only models of cardinality ℵ1 are considered. Some results are therefore connected with the Continuum Hypothesis.

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