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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

REFERENCES

[B] Bartoszyński, T., Additivity of measure implies additivity of category, Transactions of the American Mathematical Society, vol. 281 (1984), pp. 209–213.CrossRefGoogle Scholar
[K1] Kunen, K., Set theory, North-Holland, Amsterdam, 1980.Google Scholar
[K2] Kunen, K., Random and Cohen reals, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 887–911.Google Scholar
[L] Laver, R., On the consistency of Borel's conjecture, Acta Mathematica, vol. 131 (1976), pp. 151–169.Google Scholar
[MS] Martin, D. A. and Solovay, R. M., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143–178.CrossRefGoogle Scholar
[M1] Miller, A. W., Some properties of measure and category, Transactions of the American Mathematical Society, vol. 266 (1981), pp. 93–114.CrossRefGoogle Scholar
[M2] Miller, A. W., Additivity of measure implies dominating reals, Proceedings of the American Mathematical Society, vol. 91 (1984), pp. 111–117.CrossRefGoogle Scholar
[RS] Raisonnier, J. and Stern, J., The strength of measurability hypotheses, Israel Journal of Mathematics, vol. 50 (1985), pp. 337–349.CrossRefGoogle Scholar
[R] Rothberger, F., Eine Äquivalenz zwischen der Kontinuumhypothese und der Existenz der Lusinschen and Sierpińskischen Mengen, Fundamenta Mathematicae, vol. 30 (1938), pp. 215–217.CrossRefGoogle Scholar
[SH1] Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1984.Google Scholar
[SH2] Shelah, S., On cardinal invariants of the continuum, Axiomatic set theory, Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 183–207.CrossRefGoogle Scholar
[SH3] Shelah, S., Proper and improper forcing (in preparation).Google Scholar