Abstract We consider of constructing normal ultrafilters in extensions are here Easton support iterations of Prikry-type forcing notions. New ways presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.

Abstract (1) We show that it is possible to add $\kappa ^+$ -Cohen subsets to $\kappa $ with a Prikry forcing over $\kappa $ . This answers a question from [9]. (2) A strengthening of non-Galvin property is introduced. It is shown to be consistent using a single measurable cardinal which improves a previous result by S. Garti, S. Shelah, and the first author [5]. (3) A situation with Extender-based Prikry forcings is examined. This relates to a question of H. Woodin.

We continue the study started in [2] and characterize j ↾ V , where j : V [ G ] → M [ H ] is an ultrapower embedding by a normal ultrafilter after a non-stationary support iteration of Prikry forcings.

Let P be a forcing notion and $G\subseteq P$ its generic subset. Suppose that we have in $V[G]$ a $\kappa{-}$ complete ultrafilter1,2W over $\kappa $ . Set $U=W\cap V$ .

Abstract A new method for blowing up the power of a singular cardinal is presented. It allows to blow up the power of a singular in the core model cardinal of uncountable cofinality. The method makes use of overlapping extenders.

Abstract Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model where all regular uncountable cardinals are measurable in HOD is constructed.

Abstract Letκ, λ be regular uncountable cardinals such that λ >κ+is not a successor of a singular cardinal of low cofinality. We construct a generic extension withs(κ) = λ starting from a ground model in whicho(κ) = λ and prove that assuming ¬0¶,s(κ) = λ implies thato(κ) ≥ λ in the core model.

Abstract It is shown that under GCH every poset preserves its cofinality in any cofinality preserving extension. On the other hand, starting with ω measurable cardinals, a model with a partial ordered set which can change its cofinality in a cofinality preserving extension is constructed.

Abstract We prove that if μ is a regular cardinal and ℙ is a μ-centered forcing poset, then ℙ forces that (I[μ++[)V generates I[μ++] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.

Abstract We construct a model with an indecisive precipitous ideal and a model with a precipitous ideal with a non precipitous normal ideal below it. Such kind of examples were previously given by M. Foreman [2] and R. Laver [4] respectively. The present examples differ in two ways: first- they use only a measurable cardinal and second- the ideals are over a cardinal. Also a precipitous ideal without a normal ideal below it is constructed. It is shown in addition that if there is a precipitous ideal over a cardinal κ such that • after the forcing with its positive sets the cardinality of κ remains above ℵ1 • there is no a normal precipitous ideal then there is 0†.

Abstract 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 ο(κ) = κ + 1. On the other hand, it is shown that the strength of above is at least ο(κ) = κ.

Abstract We give some general criteria, when κ-complete forcing preserves largeness properties—like κ-presaturation of normal ideals on λ (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength “NSλ is αi-preserving”, for λ > α2.

Abstract We show the consistency, relative to a supercompact cardinal, of the least measurable cardinal being both strongly compact and fully Laver indestructible. We also show the consistency, relative to a supercompact cardinal, of the least strongly compact cardinal being somewhat supercompact yet not completely supercompact and having both its strong compactness and degree of supercompactness fully Laver indestructible.

Abstract The paper is a continuation of [The SCH revisited], In § 1 we define a forcing with countably many nice systems. It is used, for example, to construct a model “GCH below κ, c f κ = ℵ0, and 2κ > κ+ω” from 0(κ) = κ+ω. In §2 we define a triangle iteration and use it to construct a model satisfying “{μ ≤ λ∣c f μ = ℵ0 and pp(μ) > λ} is countable for some λ”. The question of whether this is possible was asked by S. Shelah. In §3 a forcing for blowing the power of a singular cardinal without collapsing cardinals or adding new bounded subsets is presented. Answering a question of H. Woodin, we show that it is consistent to have “c f κ = ℵ0. GCH below κ, 2κ > κ+, and ”. In §4 a variation of the forcing of [The SCH revisited, §1] is defined. It behaves nicely in iteration processes. As an application, we sketch a construction of a model satisfying: “κ is a measurable and 2κ ≥ κ+α for some α, κ < c f α < α” starting with 0(κ) = κ+α. This answers the question from Gitik's On measurable cardinals violating the continuum hypothesis.

Suppose that I is a precipitous ideal over a cardinal κ and j is a generic embedding of I. What is the nature of j? If we assume the existence of a supercompact cardinal then, by Foreman, Magidor and Shelah [FMS], it is quite unclear where some of such j's are coming from. On the other hand, if ¬∃κ0(κ) = κ++, then, by Mitchell [Mi], the restriction of j to the core model is its iterated ultrapower by measures of it. A natural question arising here is if each iterated ultrapower of can be obtained as the restriction of a generic embedding of a precipitous ideal. Notice that there are obvious limitations. Thus the ultrapower of by a measure over λ cannot be obtained as a generic embedding by a precipitous ideal over κ ≠ λ. But if we fix κ and use iterated ultrapowers of which are based on κ, then the answer is positive. Namely a stronger statement is true: Theorem. Let τ be an ordinal and κ a measurable cardinal. There exists a generic extension V* of V so that NSℵ1 (the nonstationary ideal on ℵ1) is precipitous and, for every iterated ultrapower i of V of length ≤ τ by measures of V based on κ, there exists a stationary set forcing “the generic ultrapower restricted to V is i”. Our aim will be to prove this theorem. We assume that the reader is familiar with the paper [JMMiP] by Jech, Magidor, Mitchell and Prikry. We shall use the method of that paper for constructing precipitous ideals. Ideas of Levinski [L] for blowing up 2ℵ1 preserving precipitousness and of our own earlier paper [Gi] for linking together indiscernibles will be used also.

Namba [N] proved that the nonstationary ideal over a measurable (NSκ) cannot be κ+-saturated. Baumgartner, Taylor and Wagon [BTW] asked if it is possible for NSκ to be precipitous over a measurable κ. A model with this property was constructed by the author, and shortly after Foreman, Magidor and Shelah [FMS] proved a general theorem that after collapsing of a supercompact or even a superstrong to the successor of κ, NSκ became precipitous. This theorem implies that it is possible to have the nonstationary ideal precipitous over even a supercompact cardinal. Just start with a supercompact κ and a superstrong λ > κ. Make supercompactness of κ indistractible as in [L] and then collapse λ to be κ+. The aim of our paper is to show that the existence of a supercompact cardinal alone already implies the consistency of the nonstationary ideal precipitous over a supercompact. The proof gives also the following: if κ is a λ-supercompact for λ ≥ (2κ)+, then there exists a generic extension in which κ is λ-supercompact and NSκ is precipitous. Thus, for a model with NSκ precipitous over a measurable we need a (2κ)+-supercompact cardinal κ. Jech [J] proved that the precipitous of NSκ over a measurable κ implies the existence of an inner model with o(κ) = κ+ + 1. In §3 we improve this result a little by showing that the above assumption implies an inner model with a repeat point. The paper is organized as follows. In §1 some preliminary facts are proved. The model with NSκ precipitous over a supercompact is constructed in §2.

Abstract Assuming the existence of a supercompact cardinal, we construct a model of ZFC + (There exists a nonsplitting stationary subset of ). Answering a question of Uri Abraham [A], [A-S], we prove that adding a real to the world always makes stationary

Abstract It is proven that the following statement: “there exists a club C ⊆ κ such that every α ∈ C is an inaccessible cardinal in L and, for every δ a limit point of C, C ∩ δ is almost contained in every club of δ of L” is equiconsistent with a weakly compact cardinal if δ = ℵ1, and with a weakly compact cardinal of order 1 if δ = ℵ2.