Related papers: Changing measurable into small accessible cardinal…

Assuming $\kappa$ is a supercompact cardinal and $\lambda$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $\kappa=\aleph_\omega$ and $\lambda=\aleph_{\omega+1}.$

Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer…

We produce a forcing extension of the constructible universe $\bL$ in which every universally measurable set of reals is $\uTDelta^{1}_{2}$, partially answering question CG from David Fremlin's problem list. The analogous result for…

We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb…

Using GCH, we force the following: There are continuum many simple cardinal characteristics with pairwise different values.

Cardinal characteristics of the continuum represent the boundaries in size between the countable and the continuum with respect to certain properties of sets. They are often defined as the minimum sizes of families of reals that meet some…

This paper defines a Mitchell rank for supercompact cardinals. If $\kappa$ is a $\theta$-supercompact cardinal then $o_{\theta-sc}(\kappa) = \sup \{ o_{\theta-sc}(\mu) + 1 \ | \ \mu \in m(\kappa)\}$, where $m(\kappa)$ is the collection of…

We give an alternative presentation of the ordinal notation at the strength of $\Pi^1_1-CA_0$ which allows the "uncountable" notation $\Omega$ to be interpreted "polymorphically" - that is, we allow the notation to be interpreted as…

Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we…

Assume $\lambda$ is a singular limit of $\eta$ supercompact cardinals, where $\eta \leq \lambda$ is a limit ordinal. We present two forcing methods for making $\lambda^+$ the successor of the limit of the first $\eta$ measurable cardinals…

We present two ways in which the model $L({\mathbb R})$ is canonical assuming the existence of large cardinals. We show that the theory of this model, with {\em ordinal} parameters, cannot be changed by small forcing; we show further that a…

Given an uncountable regular cardinal $\kappa$, a partial order is $\kappa$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $\kappa$ is stationary in $\mathcal{P}_\kappa(\mathbb{P})$. We…

The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing…

We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly…

We present a direct construction of stationary set preserving forcings that make $\omega$-cofinal all the members of some arbitrary set $\mathcal{K}$ of regular cardinals $\kappa > \omega_1$. In addition, it is made possible to ensure that…

Krueger showed that PFA implies that for all regular $\Theta \ge \aleph_2$, there are stationarily many $[H(\Theta)]^{\aleph_1}$ that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model…

We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located…

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and…

Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal $\kappa$…

We show that the Proper Forcing Axiom for forcing notions of size $\aleph_1$ is consistent with the continuum being arbitrarily large. In fact, assuming $GCH$ holds and $\kappa\geq\omega_2$ is a regular cardinal, we prove that there is a…