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The first-order structure of weakly Dedekind-finite set

Published online by Cambridge University Press:  12 March 2014

A. C. Walczak-Typke
A. C. Walczak-Typke*
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Wahringer Straße 25, A-1090 Wien, Austria, E-mail: agatha@logic.univie.ac.at
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Abstract

We show that infinite sets whose power-sets are Dedekind-finite can only carry ℵ0-categorical first order structures. We identify other subclasses of this class of Dedekind-finite sets, and discuss their possible first order structures.

Keywords

axiom of choiceℵ0-categorical structuresDedekind-finite
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 4 , December 2005 , pp. 1161 - 1170
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1]Baur, W., Cherlin, G., and Macintyre, A, Totally categorical groups and rings, Journal of Algebra, vol. 57 (1979), pp. 407–440.CrossRefGoogle Scholar
[2]Cherlin, G., Harrington, L., and Lachlan, A. H., ℵ0-categorical, ℵ0-stable structures, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 103–135.CrossRefGoogle Scholar
[3]Creed, P. and Truss, J. K., On o-amorphous sets, Annals of Pure and Applied Logic, vol. 101 (2000), pp. 185–226.CrossRefGoogle Scholar
[4]Degen, J. W., Some aspects and examples of infinity notions, Mathematical Logic Quarterly, vol. 40 (1994), pp. 111–124.CrossRefGoogle Scholar
[5]Felgner, U., ℵ0-categorical stable groups, Mathematische Zeitschrift, vol. 160 (1978), pp. 27–49.CrossRefGoogle Scholar
[6]Herwig, B, Macpherson, H. D., Martin, G., Nurtazin, A., and Truss, J. K., On ℵ-categorical weakly o-minimal structures., Annals of Pure and Applied Logic, vol. 101 (2000), no. 1, pp. 65–93.CrossRefGoogle Scholar
[7]Hodges, W., Model theory, Encyclopedia of mathematics and its applications, vol. 42, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[8]Lévy, A., The independence of various definitions of finiteness, Fundamenta Mathematicae, vol. 46 (1958), pp. 1–13.CrossRefGoogle Scholar
[9]Mendick, G. S. and Truss, J. K., A notion of rank in set theory without choice, Archive for Mathematical Logic, vol. 42 (2003), pp. 165–178.CrossRefGoogle Scholar
[10]Morley, M., Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514–538.CrossRefGoogle Scholar
[11]Plotkin, J. M., Generic emheddings, this Journal, vol. 34 (1969), pp. 388–394.Google Scholar
[12]Poizat, B., Stable groups, Mathematical Surveys and Monographs, vol. 87, American Mathematical Society, Providence, RI, 2001.CrossRefGoogle Scholar
[13]Steinhorn, C. and Pillay, A., Definable sets in ordered structures I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565–592.Google Scholar
[14]Tarski, A., Sur les ensembles finis, Fundamenta Mathematicae, vol. 6 (1924), pp. 45–95.CrossRefGoogle Scholar
[15]Truss, J. K., Classes of Dedekind finite cardinals, Fundamenta Mathematicae, vol. 84 (1974), pp. 187–208.CrossRefGoogle Scholar
[16]Truss, J. K., The structure of amorphous sets, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 191–233.CrossRefGoogle Scholar