Journal of Symbolic Logic

On Ideals of Subsets of the Plane and on Cohen Reals

Jacek Cichon and Janusz Pawlikowski

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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

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

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Lebesgue measure Baire category cardinal indices Cohen reals


Cichon, Jacek; Pawlikowski, Janusz. On Ideals of Subsets of the Plane and on Cohen Reals. J. Symbolic Logic 51 (1986), no. 3, 560--569.

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