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Cuts in hyperfinite time lines

Published online by Cambridge University Press:  12 March 2014

Renling Jin*
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Abstract

In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperinlegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ϵ O there is a y greater than all the elements in U such that the interval [xy, x + y] ⊆ O. Let U be a cut in a hyperfinite time line , which is a hyperfinite initial segment of the hyperintegers. U is called a good cut if there exists a U-meager subset of of Loeb measure one. Otherwise U is bad. In this paper we discuss the questions of Keisler and Leth about the existence of bad cuts and related cuts. We show that assuming b > ω1, every hyperfinite time line has a cut with both cofinality and coinitiality uncountable. We construct bad cuts in a nonstandard universe under ZFC. We also give two results about the existence of other kinds of cuts.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[Ca]Canjar, M., Countable ultraproducts without CH, Annals of Pure and Applied Logic, vol. 37 (1988), pp. 179.CrossRefGoogle Scholar
[CK]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973; 3rd ed., 1990.Google Scholar
[KKML]Keisler, H. J., Kunen, K., Miller, A. W., and Leth, S., Descriptive set theory over hyperfinite sets, this Journal, vol. 54 (1989), pp. 11671180.Google Scholar
[KL]Keisler, H. J. and Leth, S., Meager sets on the hyperfinite time line, this Journal, vol. 56 (1991), pp. 71102.Google Scholar
[Lo]Loeb, P., Conversion from nonstandard to standard measure space and applications in probability theory, Transactions of the American Mathematical Society, vol. 211 (1975), pp. 113122.CrossRefGoogle Scholar
[SB]Stroyan, K. D. and Bayod, J. M., Foundations of infinitesimal stochastic analysis, North-Holland, Amsterdam, 1986.Google Scholar