Ramsey Sets, the Ramsey Ideal, and Other Classes Over $\mathbf{R}$

Journal of Symbolic Logic 57 (4):1441-1468 (1992)
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Abstract

We improve results of Marczewski, Frankiewicz, Brown, and others comparing the $\sigma$-ideals of measure zero, meager, Marczewski measure zero, and completely Ramsey null sets; in particular, we remove CH from the hypothesis of many of Brown's constructions of sets lying in some of these ideals but not in others. We improve upon work of Marczewski by constructing, without CH, a nonmeasurable Marczewski measure zero set lacking the property of Baire. We extend our analysis of $\sigma$-ideals to include the completely Ramsey null sets relative to a Ramsey ultrafilter and obtain all 32 possible examples of sets in some ideals and not others, some under the assumption of MA, but most in ZFC alone. We also improve upon the known constructions of a Marczewski measure zero set which is not Ramsey by using a set theoretic hypothesis which is weaker than those used by other authors. We give several consistency proofs: one concerning the relative sizes of the covering numbers for the meager sets and the completely Ramsey null sets; another concerning the size of non; and a third concerning the size of add$. We also study those classes of perfect sets which are bases for the class of always first category sets.

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References found in this work

Happy families.A. R. D. Mathias - 1977 - Annals of Mathematical Logic 12 (1):59.
Iterated perfect-set forcing.James E. Baumgartner & Richard Laver - 1979 - Annals of Mathematical Logic 17 (3):271-288.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
Descriptive Set Theory.Yiannis Nicholas Moschovakis - 1982 - Studia Logica 41 (4):429-430.

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