## Journal of Symbolic Logic

### Cuts in Hyperfinite Time Lines

Renling Jin

#### Abstract

In an $\omega_1$-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 $x \in O$ 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 $\mathscr{H}$, which is a hyperfinite initial segment of the hyperintegers. $U$ is called a good cut if there exists a $U$-meager subset of $\mathscr{H}$ 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.

#### Article information

Source
J. Symbolic Logic, Volume 57, Issue 2 (1992), 522-527.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743971

Mathematical Reviews number (MathSciNet)
MR1169188

Zentralblatt MATH identifier
0770.03022

JSTOR