Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-31T19:13:25.533Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparing notions of similarity for uncountable models

Published online by Cambridge University Press:  12 March 2014

Taneli Huuskonen
Taneli Huuskonen*
Affiliation:
Department of Mathematics, P.O. Box 4 (Hallituskatu 15), FI-00014 University of Helsinki, Finland, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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-Fraïssé-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.

Keywords

Primary 03C3003C75Secondary 03C5503E40
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 4 , December 1995 , pp. 1153 - 1167
Copyright
Copyright © Association for Symbolic Logic 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[BHK]Baumgartner, J. E., Harrington, L. A., and Kleinberg, E. M., Adding a closed unbounded subset, this Journal, vol. 41 (1976), pp. 481–482.Google Scholar
[Di]Dickmann, M. A., Large infinitary languages – Model theory, North-Holland, Amsterdam/Oxford, 1975.Google Scholar
[Hul]Huuskonen, T., Comparing notions of similarity for uncountable models, academic dissertation, University of Helsinki, Department of Mathematics, 1991.Google Scholar
[Hu2]Huuskonen, T., Observations about Scott and Karp trees, revised version of part of [Hu1], Annals of Pure and Applied Logic (to appear).Google Scholar
[HV]Hyttinen, T. and Väänänen, J., On Scott and Karp trees of uncountable models, this Journal, vol. 55 (1990), pp. 897–908.Google Scholar
[Hy]Hyttinen, T., Games and Infinitary Languages, Annales Academiae Scientiarum Fennicae. Series A I. Mathematica Dissertationes, vol. 64, Helsinki, 1987.Google Scholar
[Jc]Jech, T., Set Theory, Academic Press, 1978.Google Scholar
[Mo]Moschovakis, Y. N., Descriptive Set Theory, North-Holland, Amsterdam/New York/Oxford, 1980.Google Scholar
[NS]Nadel, M. and Stavi, J., L∞λ-equivalence, isomorphism and potential isomorphism, Transactions of the American Mathematical Society, vol. 236 (1978), pp. 51–74.Google Scholar
[Tuu]Tuuri, H. M. T., Constructing Ehrenfeucht-Fraïssé-equivalent nonisomorphic models, Infinitary Languages and Ehrenfeucht-Fraïssé-Games, academic dissertation, University of Helsinki, Department of Mathematics, 1990.Google Scholar