𝑄-sets, Sierpiński sets, and rapid filters

Judah, Haim; Shelah, Saharon

doi:10.1090/S0002-9939-1991-1045594-5

$Q$-sets, Sierpiński sets, and rapid filters
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by Haim Judah and Saharon Shelah

Proc. Amer. Math. Soc. 111 (1991), 821-832

DOI: https://doi.org/10.1090/S0002-9939-1991-1045594-5

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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].

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