Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T05:48:27.207Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

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

[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