Flat sets

Journal of Symbolic Logic 59 (3):1012-1021 (1994)
Abstract
Let X be a set, and let $\hat{X} = \bigcup^\infty_{n = 0} X_n$ be the superstructure of X, where X 0 = X and X n + 1 = X n ∪ P(X n ) (P(X) is the power set of X) for n ∈ ω. The set X is called a flat set if and only if $X \neq \varnothing.\varnothing \not\in X.x \cap \hat X = \varnothing$ for each x ∈ X, and $x \cap \hat{y} = \varnothing$ for x.y ∈ X such that x ≠ y, where $\hat{y} = \bigcup^\infty_{n = 0} y_n$ is the superstructure of y. In this article, it is shown that there exists a bijection of any nonempty set onto a flat set. Also, if W̃ is an ultrapower of X̂ (generated by any infinite set I and any nonprincipal ultrafilter on I), it is shown that W̃ is a nonstandard model of X: i.e., the Transfer Principle holds for X̂ and W̃, if X is a flat set. Indeed, it is obvious that W̃ is not a nonstandard model of X when X is an infinite ordinal number. The construction of flat sets only requires the ZF axioms of set theory. Therefore, the assumption that X is a set of individuals (i.e., $x \neq \varnothing$ and a ∈ x does not hold for x ∈ X and for any element a) is not needed for W̃ to be a nonstandard model of X
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/2275925
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 38,113
Through your library

References found in this work BETA

No references found.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

U-Lusin Sets in Hyperfinite Time Lines.Renling Jin - 1992 - Journal of Symbolic Logic 57 (2):528-533.
Almost Weakly 2-Generic Sets.Stephen A. Fenner - 1994 - Journal of Symbolic Logic 59 (3):868-887.
Set Theoretic Properties of Loeb Measure.Arnold W. Miller - 1990 - Journal of Symbolic Logic 55 (3):1022-1036.
On the Kleene Degrees of Π11 Sets.Theodore A. Slaman - 1986 - Journal of Symbolic Logic 51 (2):352 - 359.
Flat Morley Sequences.Ludomir Newelski - 1999 - Journal of Symbolic Logic 64 (3):1261-1279.
Nonstandard Set Theory.Peter Fletcher - 1989 - Journal of Symbolic Logic 54 (3):1000-1008.
Σ1-Separation.Fred G. Abramson - 1979 - Journal of Symbolic Logic 44 (3):374 - 382.

Analytics

Added to PP index
2009-01-28

Total views
177 ( #35,327 of 2,313,601 )

Recent downloads (6 months)
2 ( #392,244 of 2,313,601 )

How can I increase my downloads?

Monthly downloads

My notes

Sign in to use this feature