Related papers: The Lottery Preparation

We show that if the existence of a supercompact cardinal $\kappa$ with a weakly compact cardinal $\lambda$ above $\kappa$ is consistent, then the following are consistent as well (where $\mathfrak{t}(\kappa)$ and $\mathfrak{u}(\kappa)$ are…

We analyze the intermediate models of the strongly compact Prikry forcing. We exhibit a simple combinatorial property which, for a given supercompact cardinal $\kappa$, characterize the projections of all projections of the strongly compact…

We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of…

We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$. In addition, we prove several global results…

A cardinal kappa is countably closed if mu^omega < kappa whenever mu < kappa. Assume that there is no inner model with a Woodin cardinal and that every set has a sharp. Let K be the core model. Assume that kappa is a countably closed…

Assume $\kappa = \kappa^{< \kappa}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{\kappa>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this,…

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,…

Starting from a model with a Laver-indestructible supercompact cardinal $\kappa$, we construct a model of $ZF+DC_{\kappa}$ where there are no $\kappa$-mad families.

This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new…

Let $\kappa$ be an infinite cardinal. Then, forcing with $\mathbb{R}(\kappa)$$\times$$\mathbb{R}(\kappa)$ adds a generic filter for $\mathbb{C}(\kappa);$ where $\mathbb{R}(\kappa)$ and $\mathbb{C}(\kappa)$ are the forcing notions for adding…

We show that higher Sacks forcing at a regular limit cardinal and club Miller forcing at an uncountable regular cardinal both add a diamond sequence. We answer the longstanding question, whether $\kappa = \kappa^{<\kappa} \geq\aleph_1$…

The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that…

This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's…

Vladimir Kanovei \cite{zbMATH01335192} developed the technique of geometric iteration and used it to prove that the perfect set forcing can be iterated with countable supports along any partial order, while preserving $\aleph_1$. In…

Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $\lambda>\kappa$ to $\kappa$) such that the composition still…

We study which $\kappa$-distributive forcing notions of size $\kappa$ can be embedded into tree Prikry forcing notions with $\kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the…

Given a cardinal $\kappa$ that is $\lambda$-supercompact for some regular cardinal $\lambda\geq\kappa$ and assuming $\GCH$, we show that one can force the continuum function to agree with any function $F:[\kappa,\lambda]\cap\REG\to\CARD$…

Many of the most common reverse Easton iterations found in the large cardinal context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as P*Q, where P has size less than delta and Q is forced to…

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $\kappa$ and a cardinal $\lambda$, we say that $\kappa$ has the $\lambda$-gluing property if every sequence of $\lambda$-many…

An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa.$ We prove that if there is a model of $\ZFC$ with two supercompact cardinals, then there is a model of \ZFC where…