Related papers: On Cohen and Prikry Forcing Notions

We prove it consistent relative to ZFC that all nontrivial forcings of size $\aleph _1$ add a Cohen real.

If T is an iteration tree on K and F is a countably certified extender that coheres with the final model of T, then F is on the extender sequence of the final model of T. Several applications of maximality are proved, including: o K…

Given a weakly compact cardinal $\kappa$, we give an axiomatization of intuitionistic first-order logic over $\mathcal{L}_{\kappa^+, \kappa}$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the…

We prove that superclub implies $\mathfrak{s}=\aleph_1$. More generally, superclub at a successor of a weakly compact cardinal implies $\mathfrak{s}_\kappa=\kappa^+$. Based on this statement, we separate tiltan from superclub at a successor…

We prove a theorem on iterated forcing that can be used for preservation of $\aleph_2$ and $\aleph_1$ in iterations with supports of size $\aleph_1$ of forcings that have amalgamation properties similar to those present in the perfect set…

In chapter 9 of his book "The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal", Woodin shows how to force the Strong Chang Conjecture over models of determinacy using $\mathbb{P}_{\mathrm{max}}$. We show here how a…

The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a…

In this paper we first formulate several ``combinatorial principles'' concerning kappa \times omega matrices of subsets of omega and prove that they are valid in the generic extension obtained by adding any number of Cohen reals to any…

In this paper we produce models $V_1\subseteq V_2$ of set theory such that adding $\kappa$-many Cohen reals to $V_2$ adds $\lambda$-many Cohen reals to $V_1$, for some $\lambda>\kappa$. We deal mainly with the case when $V_1$ and $V_2$ have…

We show that for any regular cardinal $\kappa$, $\square_{\kappa, 2}$ is consistent with "all $\kappa^+$-Aronszajn trees are special." By a result of Shelah and Stanley this is optimal in the sense that $\square_{\kappa, 2}$ may not be…

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…

Let $\mathcal{SN}$ be the $\sigma$-ideal of the strong measure zero sets of reals. We present general properties of forcing notions that allow to control of the additivity of $\mathcal{SN}$ after finite support iterations. This is applied…

The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $\lambda\leq \kappa$, the following theories are equiconsistent modulo ZFC:…

Shelah considered a certain version of Strong Chang's Conjecture, which we denote $\text{SCC}^{\text{cof}}$, and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an…

In the context of large cardinals, the classical diamond principle Diamond_kappa is easily strengthened in natural ways. When kappa is a measurable cardinal, for example, one might ask that a Diamond_kappa sequence anticipate every subset…

We continue Gitik, Kanovei and Koepke's work and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.

In this paper, we prove that: if $\kappa$ is supercompact and the $\mathsf{HOD}$ Hypothesis holds, then there is a proper class of regular cardinals in $V_{\kappa}$ which are measurable in $\mathsf{HOD}$. Woodin also proved this result. As…

Several variants of the Halpern-L\"auchli Theorem for trees of uncountable height are investigated. For $\kappa$ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one…

We study saturation properties of $\sigma$-complete measures on $P_\kappa(\lambda)$, where $\lambda$ can be either regular or singular. In particular, we prove that in contrast to Galvin's theorem, the Galvin property of…

Given an uncountable cardinal $\kappa$, we consider the question of whether subsets of the power set of $\kappa$ that are usually constructed with the help of the Axiom of Choice are definable by $\Sigma_1$-formulas that only use the…