MORE ON REAL-VALUED MEASURABLE CARDINALS AND FORCING WITH IDEALS by Moti Gitik

(1) It is shown that if c is real-valued measurable then the Maharam type of (c,P(c), σ) is 2. This answers a question of D. Fremlin [Fr,(P2f)]. (2) A different construction of a model with a real-valued measurable cardinal is given from that of R. Solovay [So]. This answers a question of D. Fremlin [Fr,(P1)]. (3) The forcing with a κ-complete ideal over a set X , |X | ≥ κ cannot be isomorphic to Random×Cohen or Cohen×Random. The result for X = κ was proved in [Gi-Sh1] but as was pointed out to us by M. Burke the application of it in [Gi-Sh2] requires dealing with any X . In Section 1 we deal with the Maharam types of real-valued measurable cardinals. The result (1) stated in the abstract and its stronger version are proved. The proofs are based on Shelah’s strong covering lemmas and his revised power set operation. In Section 2 a model with a real-valued measurable which is not obtained as the Solovay one by forcing random reals over a model with a measurable. In Section 3, the result (3) stated in the abstract is proved. Theorem 1.1 and the construction of Section 2 is due to the first author. Theorem 1.2 is joint and the result of Section 3 is due to the second author. We are grateful to David Fremlin for bringing the questions on real-valued measurability to our attention. His excellent survey article [Fr] gave the inspiration for the present paper. We wish to thank the Max Burke for pointing out a missing stage in the argument of [Gi-Sh2]. 1. On Number of Cohen or Random Reals D. Fremlin asked the following in [Fr,(P2f)]: If c is a real-valued measurable with witnessing probability ν, does it follow that the Maharam type of (c,P(c), ν) is 2? or in equivalent formulation: If c is a real-valued measurable does the forcing with witnessing ideal isomorphic to the forcing for adding 2 random reals? The next theorem provides the affirmative answer. Theorem 1.1. Suppose that I is a 20 -complete ideal over 20 and the forcing with it (i.e. P ( 20 ) /I) is isomorphic to the adding of λ-Cohen or λ-random reals. Then λ = 2 א0 . Proof: Suppose otherwise. Denote 20 by κ. Let j : V → N be a generic elementary embedding. Claim 1. j(κ) > (λ) . Proof: By a theorem of Prikry [Pr] (see also [Gi-Sh2] for a generalization) for every τ < κ 2 = 20 = κ. Then, in N , 2 = j(κ). But (P(κ)) ⊆ N , so j(κ) ≥ (2) . By [Gi-Sh2], then (2) = cov(λ, κ,א1, 2. So cov(λ, κ,א1, 2) ≥ λ . Clearly, cov(λ, κ,א1, 2) ≤ cov(λ,א1,א1, 2) ≤ (cov(λ,א1,א1, 2)) N .