## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 55, Issue 3 (1990), 909-927.

### The Kunen-Miller Chart (Lebesgue Measure, The Baire Property, Laver Reals and Preservation Theorems for Forcing)

#### Abstract

In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in \S1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We prove a preservation theorem for countable-support forcing notions, and using this theorem we prove (iii) If we add $\omega_2$ Laver reals, then the old reals have outer measure one. From this we obtain (iv) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg B(m) + \neg U(m) + U(c))$. In \S2: (i) We prove a preservation theorem, for the finite support forcing notion, of the property "$F \subseteq ^\omega\omega$ is an unbounded family." (ii) We introduce a new forcing notion making the old reals a meager set but the old members of $^\omega\omega$ remain an unbounded family. Using this we prove (iii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + U(m) + \neg B(c) + \neg U(c) + C(c))$. In \S3: (i) We prove a preservation theorem, for the finite support forcing notion, of a property which implies "the union of the old measure zero sets is not a measure zero set," and using this theorem we prove (ii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg U(m) + C(m) + \neg C(c))$.

#### Article information

**Source**

J. Symbolic Logic, Volume 55, Issue 3 (1990), 909-927.

**Dates**

First available in Project Euclid: 6 July 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1183743396

**Mathematical Reviews number (MathSciNet)**

MR1071305

**Zentralblatt MATH identifier**

0718.03037

**JSTOR**

links.jstor.org

#### Citation

Judah, Haim; Shelah, Saharon. The Kunen-Miller Chart (Lebesgue Measure, The Baire Property, Laver Reals and Preservation Theorems for Forcing). J. Symbolic Logic 55 (1990), no. 3, 909--927. https://projecteuclid.org/euclid.jsl/1183743396