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Nonstandard natural number systems and nonstandard models

Published online by Cambridge University Press:  12 March 2014

Shizuo Kamo
Shizuo Kamo*
Affiliation:
University of Osaka Prefecture, Sakai, Osaka, Japan
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Abstract

It is known (see [1, 3.1.5]) that the order type of the nonstandard natural number system *N has the form ω + (ω* + ω) θ, where θ is a dense order type without first or last element and ω is the order type of N. Concerning this, Zakon [2] examined *N more closely and investigated the nonstandard real number system *R, as an ordered set, as an additive group and as a uniform space. He raised five questions which remained unsolved. These questions are concerned with the cofinality and coinitiality of θ (which depend on the underlying nonstandard universe *U). In this paper, we shall treat nonstandard models where the cofinality and coinitiality of θ coincide with some appropriated cardinals. Using these nonstandard models, we shall give answers to three of these questions and partial answers to the other to questions in [2].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 2 , June 1981 , pp. 365 - 376
Copyright
Copyright © Association for Symbolic Logic 1981

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References

REFERENCES

[1]Robinson, A., Non-standard analysis, North-Holland, Amsterdam, 1966.Google Scholar
[2]Zakon, E., Remark on the nonstandard real axis, Applications of model theory to algebra, analysis, and probability, (Luxemburg, W. A. J., Editor), Holt, Rinehart and Winston, 1969, pp. 195–227.Google Scholar
[3]Machover, M. and Hirschfeld, J., Lecture in non-standard analysis, Springer-Verlag, 1969.CrossRefGoogle Scholar
[4]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[5]Sacks, G., Saturated model theory, Benjamin, Reading, Massachusetts, 1972.Google Scholar