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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)
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)
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)