# The Kunen-Miller chart (lebesgue measure, the baire property, Laver reals and preservation theorems for forcing)

Journal of Symbolic Logic 55 (3):909-927 (1990)
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 ω 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 ω ω 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) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg U(m) + C(m) + \neg C(c))$
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/2274464
Options
 Save to my reading list Follow the author(s) My bibliography Export citation Find it on Scholar Edit this record Mark as duplicate Revision history Request removal from index

 PhilPapers Archive Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 20,903 External links Setup an account with your affiliations in order to access resources via your University's proxy server Configure custom proxy (use this if your affiliation does not provide a proxy) Through your library Sign in / register and configure your affiliation(s) to use this tool.Configure custom resolver
References found in this work BETA

No references found.

Citations of this work BETA
Tomek Bartoszynski & Saharon Shelah (1992). Closed Measure Zero Sets. Annals of Pure and Applied Logic 58 (2):93-110.
Similar books and articles

2009-01-28

8 ( #394,359 of 1,907,521 )