RVM, RVC revisited: Clubs and Lusin sets

by Kumar and Shelah. [KmSh:1046]

A cardinal kappa is Cohen measurable (RVC) if for some kappa-additive ideal {I} over kappa, {P}(kappa) slash {I} is forcing isomorphic to adding lambda Cohen reals for some lambda . Such cardinals can be obtained by starting with a measurable cardinal kappa and adding at least kappa Cohen reals. We construct various models of RVC with different properties than this model.Our main results are: (1) kappa = 2^{omega} is RVC does not decide club_S for various stationary S subseteq kappa . (2) kappa <= lambda = cf(lambda) < 2^{omega} does not decide club_S for various stationary S subseteq lambda . (3) kappa = 2^{omega} is RVC does not decide the existence of a Lusin set of size kappa .


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