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Encoding orders and trees in binary relations

Part of: Model theory

Published online by Cambridge University Press:  26 February 2010

Wilfrid Hodges
Wilfrid Hodges
Affiliation:
Department of Mathematics, Bedford College, Regent's Park, London NW1 4NS.
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Extract

In Section 1 below I describe two measures of the complexity of a binary relation. J The theorem says that these two measures never disagree very much. Both measures of complexity arose in connection with Saharon Shelah's notion [5] of a stable firstt order theory; Shelah showed in effect that one measure is finite, if, and only if, the other is finite too. This follows trivially from the theorem below. I confess my main aim was not to get the extra information which the theorem provides, but to eliminate Shelah's use of uncountable cardinals, which seemed strangely heavy machinery for proving a purely finitary result. Section 2 below explains the modeltheoretic setting.

MSC classification

Secondary: 03C05: Equational classes, universal algebra
Type
Research Article
Information
Mathematika , Volume 28 , Issue 1 , June 1981 , pp. 67 - 71
Copyright
Copyright © University College London 1981

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References

1.Baldwin, J. T. and Lachlan, A. H.. “On universal Horn classes categorical in some infinite power”, Algebra Universalis, 3 (1973), 98–111.CrossRefGoogle Scholar
2.Baur, Walter. “No-categorical modules”, J. Symbolic Logic, 40 (1975), 213–220.CrossRefGoogle Scholar
3.Baur, Walter. “Elimination of quantifiers for modules”, Israel J. Math., 25 (1976), 64–70.CrossRefGoogle Scholar
4.Erdös, P. and Makkai, M.. “Some remarks on set theory, X”, Studia Scient. Math. Hungarica, 1 (1966), 157–159.Google Scholar
5.Shelah, S.. Classification theory (North-Holland, Amsterdam 1978).Google Scholar