Related papers: More about lambda-support iterations of <lambda-co…
This article continues Roslanowski and Shelah math.LO/9906024 and 1105.6049 We introduce here yet another property of (<lambda)-strategically complete forcing notions which implies that their lambda-support iterations do not collapse…
We look for a parallel to the notion of ``proper forcing'' among lambda-complete forcing notions not collapsing lambda^+ . We suggest such a definition and prove that it is preserved by suitable iterations.
We introduce several properties of forcing notions which imply that their lambda-support iterations are lambda-proper. Our methods and techniques refine those studied in math.LO/9906024, math.LO/0210205, math.LO/0508272 and math.LO/0605067,…
For $\lambda$ inaccessible, we may consider $(< \lambda)$-support iteration of some specific $(<\lambda)$-complete $\lambda^+$-c.c. forcing notion. But this fails a "preservation by restricting to a sub-sequence of the forcing, we "correct"…
We introduce more properties of forcing notions which imply that their lambda-support iterations are lambda-proper, where lambda is an inaccessible cardinal. This paper is a direct continuation of section A.2 of math.LO/0210205. As an…
We consider $(<\lambda)$-support iterations of a version of $(<\lambda)$-strategically complete $\lambda^+$-c.c. definable forcing notions along partial orders. We show that such iterations can be corrected to yield an analog of a result by…
We investigate the problem of when $\leq\lambda$--support iterations of $<\lambda$--complete notions of forcing preserve $\lambda^+$. We isolate a property -- {\em properness over diamonds} -- that implies $\lambda^+$ is preserved and show…
The preservation theorems for semi-properness, hemi-properness, and pseudo-completeness hold for countable support iterations as well as revised countable support iterations, notwithstanding the fact that the "factor lemma" fails for the…
We deal with (< kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues…
We give an exposition of an iteration theorem for iterating $(<\lambda)$-closed stationary $\lambda^+$-cc forcing with supports of size $<\lambda$ and preserving these two properties. We discuss the relation of this theorem with other…
We continue investigations of reasonable ultrafilters on uncountable cardinals defined in math.LO/0407498. We introduce stronger properties of ultrafilters and we show that those properties may be handled in lambda-support iterations of…
In a self-contained way, we deal with revised countable support iterated forcing for the reals. We improve theorems on preservation of the property UP, weaker than semi proper, and we hopefully improve the presentation. We continue [Sh:b,…
We develop the theory of the forcing with trees and creatures for an inaccessible lambda continuing Ros{\l}anowski and Shelah math.LO/9807172, math.LO/9909115. To make a real use of these forcing notions (that is to iterate them without…
We prove an iteration theorem which guarantees for a wide class of nice iterations of $\omega_1$-preserving forcings that $\omega_1$ is not collapse, at the price of needing large cardinals to burn as fuel. More precisely, we show that a…
We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $\kappa$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $\kappa$. Moreover assuming GCH,…
We deal with the problem of preserving various versions of completeness in (< kappa) --support iterations of forcing notions, generalizing the case ``S --complete proper is preserved by CS iterations for a stationary co-stationary S…
A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is…
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a…
We give some general criteria, when kappa-complete forcing preserves largeness properties -- like kappa-presaturation of normal ideals on lambda (even when they concentrate on small cofinalities). Then we quite accurately obtain the…
We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are…