Related papers: On a Conjecture Regarding the Mouse Order for Weas…

We prove that for regular $\lambda$ above a strong limit singular $\mu$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $\lambda$ and $\mu$ there are coboundedly many regular…

We show that if \kappa\ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size \kappa\ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2^\kappa\…

We define a new inner model C2(omega) based on the fragment of second order logic in which second order variables range over countable subsets of the domain. We compare C2(omega) to the previously studied inner model C(aa). We argue that…

We study ultrafilters on regular uncountable cardinals, with a primary focus on $\omega_1$, and particularly in relation to the Tukey order on directed sets. Results include the independence from ZFC of the assertion that every uniform…

We develop the theory of layered posets, and use the notion of layering to prove a new iteration theorem (Theorem 6): if $\kappa$ is weakly compact then any universal Kunen iteration of $\kappa$-cc posets (each possibly of size $\kappa$) is…

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…

For which (first-order complete, usually countable) $T$ do there exist non-isomorphic models of $T$ which become isomorphic after forcing with a forcing notion $\mathbb{P}$? Necessarily, $\mathbb{P}$ is non-trivial; i.e.~it adds some new…

In this paper we introduce a new hierarchy of large cardinals between I3 and I2, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.

Usuba has asked whether the $\kappa$-mantle, the intersection of all grounds that extend to $V$ via a forcing of size ${<}\kappa$, is always a model of ZFC. We give a negative answers by constructing counterexamples where $\kappa$ is a…

In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees (see \rdef{tame iteration tree}) implies that in some homogenous generic extension of $V$ there is a transitive model $M$ containing $Ord \cup…

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 $F:[\kappa,\lambda]\cap\REG\to\CARD$…

Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal $\delta$ that is both a limit of Woodin cardinals and a limit of…

We give an example of a countable theory T such that for every cardinal lambda >= aleph_2 there is a fully indiscernible set A of power lambda such that the principal types are dense over A, yet there is no atomic model of T over A. In…

We prove the following version of the Loebl-Komlos-Sos Conjecture: For every alpha>0 there exists a number M such that for every k>M every n-vertex graph G with at least (0.5+alpha)n vertices of degree at least (1+alpha)k contains each tree…

This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity…

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…

Let M be a fine structural mouse. Let D be a fully backgrounded L[E]-construction computed inside an iterable coarse premouse S. We describe a process comparing M with D, through forming iteration trees on M and on S. We then prove that…

In this paper we consider nonmeasurablity with respect to sigma-ideals defined be trees. First classical example of such ideal is Marczewski ideal s_0. We will consider also ideal l_0 defined by Laver trees and m_0 defined by Miller trees.…

We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem: Suppose $\kappa$ is a singular strong limit cardinal and…

This article is Part I in a series of three papers devoted to determining the minimal complexity of scales in the inner model $K(\mathbb{R})$. Here, in Part I, we shall complete our development of a fine structure theory for $K(\mathbb{R})$…