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Pseudo-superstructures as nonstandard universes

Published online by Cambridge University Press:  12 March 2014

Mauro Di Nasso
Mauro Di Nasso*
Affiliation:
Dipartimento Di Matematica, Università Di Siena, Via Del Capitano, 15, 53100 Siena, Italy, E-mail: dinasso@unisi.it
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Abstract

A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed. Though necessarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values. Nonstandard universes are also constructed which model the whole ZFC theory without regularity and satisfy the κ-saturation property.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 222 - 236
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1]Aczel, P., Non-well-founded sets, CSLI Lecture Notes, no. 14, Center for the Study of Language and Information, Stanford, California, 1988.Google Scholar
[2]America, P. and Rutten, J. J. M. M., Solving domain equations in a category of complete metric spaces, Journal of Computer and Systems Sciences, vol. 39 (1989), pp. 343–375.CrossRefGoogle Scholar
[3]Ballard, D. and Hrbacek, K., Standard foundations for nonstandard analysis, this Journal, vol. 57 (1992), pp. 741–748.Google Scholar
[4]Boffa, M., Forcing et Négation de l'Axiome de Fondement, Memoirs of the Belgian Academy of Sciences, vol. XL (1972), no. 7.Google Scholar
[5]Chang, C. C. and Keisler, H. J., Model theory, 3 ed., North-Holland, Amsterdam, 1990, 1st edition 1973.Google Scholar
[6]Nasso, M. Di, Hyperordinals and nonstandard α-models, Logic and algebra (Ursini, A. and Aglianò, P., editors), Lecture Notes in Pure and Applied Mathematics, no. 180, Dekker, New York, 1996, pp. 457–475.Google Scholar
[7]Fletcher, P., Nonstandard set theory, this Journal, vol. 54 (1989), pp. 1000–1008.Google Scholar
[8]Forti, M. and Honsell, F., Set theory with free construction principles, Annali delta Scuola Normale Superiore di Pisa, Classe dt Scienze, vol. 10 (1983), pp. 493–522.Google Scholar
[9]Hinnion, R. (editor), L'Anti-fondation en Logique et en Théorie des Ensembles, Cahiers du Centre de Logique, no. 7, Louvain-la-Neuve, 1992.Google Scholar
[10]Hrbacek, K., Nonstandard set theory, The American Mathematical Monthly, vol. 86 (1979), pp. 659–677.CrossRefGoogle Scholar
[11]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[12]Kawai, T., Nonstandard analysis by axiomatic method, Southeast Asian conference on logic (Chong, C. T. and Wicks, M. J., editors), Studies in Logic and the Foundations of Mathematics, no. 111, North-Holland, Amsterdam, 1983, Singapore, 1981, pp. 55–76.Google Scholar
[13]Keisler, H. J., Foundations of infinitesimal calculus, Prindle, Weber & Schmidt, Boston, 1976.Google Scholar
[14]Nelson, E., Internal set theory: a new approach to nonstandard analysis, Bulletin of the American Mathematical Society, vol. 83 (1977), pp. 1165–1198.CrossRefGoogle Scholar
[15]Robinson, A. and Zakon, E., A set-theoretical characterization of enlargements, Applications of model theory to algebra, analysts and probability (Luxemburg, W. A. J., editor), Holt, Rinehart and Winston, New York, 1969, pp. 109–122.Google Scholar
[16]Schmid, J. and Schmidt, J., Enlargements without urelements, Colloquium Mathematicum, vol. 52 (1987), pp. 1–22.CrossRefGoogle Scholar
[17]Zakon, E., A new variant of nonstandard analysis, Victoria symposium on nonstandard analysts (Hurd, A. E. and Loeb, P. A., editors), Lecture Notes in Mathematics, no. 369, Springer-Verlag, Berlin, 1974, pp. 313–339.CrossRefGoogle Scholar