论文信息 - RVM , RVC revisited : Clubs and Lusin sets ∗

RVM , RVC revisited : Clubs and Lusin sets ∗

A cardinal κ is Cohen measurable (RVC) if for some κ-additive ideal I over κ, P(κ)/I is forcing isomorphic to adding λ Cohen reals for some λ. Such cardinals can be obtained by starting with a measurable cardinal κ and adding at least κ Cohen reals. We construct various models of RVC having different properties than this model. Our main results are: (1) κ = 2א0 is RVC does not decide ♣S for various stationary S ⊆ κ. (2) κ is RVC and κ < λ = cf(λ) < 2א0 does not decide ♣S for various stationary S ⊆ λ. (3) κ = 2א0 is RVC does not decide the existence of a Lusin set of size κ. We also prove analogues of (1), (2) for real valued measurable cardinals.

S. Shelah | Ashutosh Kumar

[1] Saharon Shelah,et al. Models of Cohen measurability , 2014, Ann. Pure Appl. Log..

[2] S. Shelah. Many partition relations below density , 2009, Israel Journal of Mathematics.

[3] Saharon Shelah,et al. Proper and Improper Forcing , 1998 .

[4] S. Shelah,et al. Sticks and Clubs , 1997, Ann. Pure Appl. Log..

[5] S. Shelah. Possibly every real function is continuous on a non--meagre set , 1995, math/9511220.

[6] Saharon Shelah,et al. Forcings with ideals and simple forcing notions , 1989 .

[7] Saharon Shelah,et al. On certain indestructibility of strong cardinals and a question of Hajnal , 1989, Arch. Math. Log..

[8] Kenneth Kunen,et al. Set Theory: An Introduction to Independence Proofs , 2010 .

[9] Robert M. Solovay,et al. Real-valued measurable cardinals , 1967 .

[10] T. J. Clogger. Diamonds , 1950 .