An old question of T. Jech and K. Prikry asks if an existence of a precipitous ideal implies necessary existence of a normal precipitous ideal. The aim of the paper is to prove some results in the positive direction. Thus, it is shown that under some mild assumptions, an existence of a precipitous ideal over ℵ 1 implies an existence of a normal precipitous… (More)
(1) It is shown that if c is real-valued measurable then the Maharam type of (c, P(c), σ) is 2 c. 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… (More)
We prove the following Theorem. Suppose M is a countable model of ZFC and k is an almost huge cardinal in M. Let A be a subset of k consisting of nonlimit ordinals. Then there is a model NA of ZF such that S0 is a regular cardinal in NA iff a e A for every a > 0. 0. Introduction. We consider the following question. What are the restrictions in ZF on the… (More)
A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with o(κ) = κ + 1. On the other hand, it is shown that the strength of above is at least o(κ) = κ.
We answer some question of [Gi]. The upper bound of [Gi] on the strength of N S µ + precipitous for a regular µ is proved to be exact. It is shown that saturatedness of N S ℵ 0 κ over inaccessible κ requires at least o(κ) = κ ++. The upper bounds on the strength of N S κ precipitous for inaccessible κ are reduced quite close to the lower bounds.
We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem. Suppose κ is a singular strong limit cardinal and 2 κ ≥ λ where λ is not the successor of a cardinal of cofinality at most κ. If cf(κ) > ω then it follows that o(κ) ≥ λ, and if… (More)