Cuts in hyperfinite time lines

Journal of Symbolic Logic 57 (2):522-527 (1992)
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Abstract

In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers 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 ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time line H, which is a hyperfinite initial segment of the hyperintegers. U is called a good cut if there exists a U-meager subset of H 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 $\mathbf{b} > \omega_1$, 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

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Citations of this work

Game sentences and ultrapowers.Renling Jin & H. Jerome Keisler - 1993 - Annals of Pure and Applied Logic 60 (3):261-274.

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References found in this work

Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
Countable ultraproducts without CH.Michael Canjar - 1988 - Annals of Pure and Applied Logic 37 (1):1-79.
Foundations of Infinitesimal Stochastic Analysis.K. D. Stroyan - 1988 - Journal of Symbolic Logic 53 (4):1261-1262.

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