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$ Q$-sets, Sierpiński sets, and rapid filters

Authors: Haim Judah and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 111 (1991), 821-832
MSC: Primary 03E35; Secondary 54A25
MathSciNet review: 1045594
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Abstract: In this work we will prove the following:

Theorem 1. cons(ZF) implies cons(ZFC + there exists a $ Q$-set of reals + there exists a set of reals of cardinality $ {\aleph _1}$, which is not Lebesgue measurable).

Theorem 2. cons(ZF) implies cons(ZFC+ $ {2^{{\aleph _0}}}$ is arbitrarily larger than $ {\aleph _2}$+ there exists a Sierpinski set of cardinality $ {2^{{\aleph _0}}}$ + there are no rapid filters on $ \omega $).

These theorems give answers to questions of Fleissner [F1] and Judah [Ju].

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1991 American Mathematical Society

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