Journal of Symbolic Logic 57 (2):528-533 (1992)

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. A subset B of H is called a U-Lusin set in H if B is uncountable and for any Loeb-Borel U-meager subset X of H, B ∩ X is countable. Here a Loeb-Borel set is an element of the σ-algebra generated by all internal subsets of H. In this paper we answer some questions of Keisler and Leth about the existence of U-Lusin sets by proving the following facts. (1) If $U = x/\mathbb{N} = \{y \in \mathscr{H}: \forall n \in \mathbb{N}(y < x/n)\}$ for some x ∈ H, then there exists a U-Lusin set of power κ if and only if there exists a Lusin set of the reals of power κ. (2) If U ≠ x/N but the coinitiality of U is ω, then there are no U-Lusin sets if CH fails. (3) Under ZFC there exists a nonstandard universe in which U-Lusin sets exist for every cut U with uncountable cofinality and coinitiality. (4) In any ω2-saturated nonstandard universe there are no U-Lusin sets for all cuts U except U = x/N
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/2275287
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 62,356
Through your library

References found in this work BETA

Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
Foundations of Infinitesimal Stochastic Analysis.K. D. Stroyan - 1988 - Journal of Symbolic Logic 53 (4):1261-1262.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Meager Sets on the Hyperfinite Time Line.H. Jerome Keisler & Steven C. Leth - 1991 - Journal of Symbolic Logic 56 (1):71-102.
U-Monad Topologies of Hyperfinite Time Lines.Renling Jin - 1992 - Journal of Symbolic Logic 57 (2):534-539.
Existence of Some Sparse Sets of Nonstandard Natural Numbers.Renling Jin - 2001 - Journal of Symbolic Logic 66 (2):959-973.
Set Theoretic Properties of Loeb Measure.Arnold W. Miller - 1990 - Journal of Symbolic Logic 55 (3):1022-1036.
Lusin-Sierpinski Index for the Internal Sets.Boško Živaljević - 1992 - Journal of Symbolic Logic 57 (1):172 - 178.
Type Two Cuts, Bad Cuts and Very Bad Cuts.Renling Jin - 1997 - Journal of Symbolic Logic 62 (4):1241-1252.
Cuts in Hyperfinite Time Lines.Renling Jin - 1992 - Journal of Symbolic Logic 57 (2):522-527.

Analytics

Added to PP index
2009-01-28

Total views
27 ( #402,629 of 2,445,362 )

Recent downloads (6 months)
1 ( #457,131 of 2,445,362 )

How can I increase my downloads?

Downloads

My notes