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
Download options
Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 25,662
Through your library
References found in this work BETA

No references found.

Add more references

Citations of this work BETA
Combinatorial Properties of Hechler Forcing.Jörg Brendle, Haim Judah & Saharon Shelah - 1992 - Annals of Pure and Applied Logic 58 (3):185-199.
Closed Measure Zero Sets.Tomek Bartoszynski & Saharon Shelah - 1992 - Annals of Pure and Applied Logic 58 (2):93-110.
Combinatorial Properties of Classical Forcing Notions.Jörg Brendle - 1995 - Annals of Pure and Applied Logic 73 (2):143-170.

View all 8 citations / Add more citations

Similar books and articles
Strong Measure Zero Sets Without Cohen Reals.Martin Goldstern, Haim Judah & Saharon Shelah - 1993 - Journal of Symbolic Logic 58 (4):1323-1341.
Solovay Models and Forcing Extensions.Joan Bagaria & Roger Bosch - 2004 - Journal of Symbolic Logic 69 (3):742-766.
Forcing Disabled.M. C. Stanley - 1992 - Journal of Symbolic Logic 57 (4):1153-1175.
Strong Measure Zero Sets and Rapid Filters.Jaime I. Ihoda - 1988 - Journal of Symbolic Logic 53 (2):393-402.
Souslin Forcing.Jaime I. Ihoda & Saharon Shelah - 1988 - Journal of Symbolic Logic 53 (4):1188-1207.
Mathias Absoluteness and the Ramsey Property.Lorenz Halbeisen & Haim Judah - 1996 - Journal of Symbolic Logic 61 (1):177-194.
Generic Trees.Otmar Spinas - 1995 - Journal of Symbolic Logic 60 (3):705-726.

Monthly downloads

Added to index

2009-01-28

Total downloads

9 ( #447,496 of 2,143,766 )

Recent downloads (6 months)

1 ( #386,855 of 2,143,766 )

How can I increase my downloads?

My notes
Sign in to use this feature


Discussion
Order:
There  are no threads in this forum
Nothing in this forum yet.

Other forums