## Journal of Symbolic Logic

### On Ideals of Subsets of the Plane and on Cohen Reals

#### 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

https://projecteuclid.org/euclid.jsl/1183742155

Mathematical Reviews number (MathSciNet)
MR853839

Zentralblatt MATH identifier
0622.03035

JSTOR