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The Kunen-Miller chart (Lebesgue measure, the Baire property, Laver reals and preservation theorems for forcing)

Published online by Cambridge University Press:  12 March 2014

Haim Judah
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720 Mathematical Sciences Research Institute, Berkeley, California 94720 Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan, Israel Institute of Mathematics, The Hebrew University, Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
Saharon Shelah
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720 Mathematical Sciences Research Institute, Berkeley, California 94720 Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan, Israel Institute of Mathematics, The Hebrew University, Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

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 §1: (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 ω2 Laver reals, then the old reals have outer measure one. From this we obtain (iv) Cons(ZF) Cons(ZFC + ¬ B(m) + ¬ U(m) + U(c)). In §2: (i) We prove a preservation theorem, for the finite support forcing notion, of the property “Fωω is an unbounded family.” (ii) We introduce a new forcing notion making the old reals a meager set but the old members of ωω remain an unbounded family. Using this we prove (iii) Cons(ZF) ⇒ Cons(ZFC + U(m) + ¬ B(c) + ¬ U(c) + C(c)). In §3: (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) Cons(ZF) ⇒ Cons(ZFC + ¬U(m) + C(m) + ¬ C(c)).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1990

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References

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The Kunen-Miller chart (Lebesgue measure, the Baire property, Laver reals and preservation theorems for forcing)
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