Journal of Symbolic Logic

On Ideals of Subsets of the Plane and on Cohen Reals

Jacek Cichon and Janusz Pawlikowski

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Abstract

Let $\mathscr{J}$ be any proper ideal of subsets of the real line $R$ which contains all finite subsets of $R$. We define an ideal $\mathscr{J}^\ast\mid\mathscr{B}$ as follows: $X \in \mathscr{J}^\ast\mid\mathscr{B}$ if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any $x \in R$ we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$. We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power $\omega_1$ such that $\bigcup\mathscr{A} \not\in \mathscr{J}^\ast\mid\mathscr{B}$. In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory.

Article information

Source
J. Symbolic Logic, Volume 51, Issue 3 (1986), 560-569.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183742155

Mathematical Reviews number (MathSciNet)
MR853839

Zentralblatt MATH identifier
0622.03035

JSTOR
links.jstor.org

Subjects
Primary: 03E35: Consistency and independence results
Secondary: 04A15

Keywords
Lebesgue measure Baire category cardinal indices Cohen reals

Citation

Cichon, Jacek; Pawlikowski, Janusz. On Ideals of Subsets of the Plane and on Cohen Reals. J. Symbolic Logic 51 (1986), no. 3, 560--569. https://projecteuclid.org/euclid.jsl/1183742155


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