Related papers: Of Mice and Machetes

Let $M$ be a fine structural mouse and let $F\in M$ be such that $M\models$``$F$ is a total extender'' and $(M||\mathrm{lh}(F),F)$ is a premouse. We show that it follows that $F\in\mathbb{E}^M$, where $\mathbb{E}^M$ is the extender sequence…

Suppose there is a Reinhardt cardinal. Then (1) $M_n(X)$ exists and is fully iterable (above $X$) for every transitive set $X$ and every $n<\omega$ (here $M_n(X)$ denotes the canonical minimal proper class inner model containing $X$ and…

We investigate Steel's conjecture in 'The Core Model Iterability Problem', that if $W$ and $R$ are $\Omega+1$-iterable, $1$-small weasels, then $W\leq^{*}R$ iff there is a club $C\subset\Omega$ such that for all $\alpha\in C$, if $\alpha$…

We show that if the universe is self-iterable and $\kappa$ is an inaccessible limit of Woodin cardinal then $AD_R + "\Theta$ is regular" holds in the derived model at $\kappa$. The proof is fine-structure free, and only assumes basic…

Let $M$ be a $\lambda$-indexed (that is, Jensen indexed) premouse. We prove that $M$ is iterable with respect to standard $\lambda$-iteration rules iff $M$ is iterable with respect to a natural version of Mitchell-Steel iteration rules.…

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 give a construction of scales (in the descriptive set theoretic sense) directly from mouse existence hypotheses, without using any determinacy arguments. The construction is related to the Martin-Solovay construction for scales on…

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible…

We give a detailed proof of the properties of the usual Prikry type forcing notion for turning a measurable cardinal into $\aleph_\omega$.

Assume ZF + AD + $V=L(\mathbb{R})$. We prove some "mouse set" theorems, for definability over $J_\alpha(\mathbb{R})$ where $[\alpha,\alpha]$ is a projective-like gap (of $L(\mathbb{R})$) and $\alpha$ is either a successor ordinal or has…

We extend and improve the result of Makkai and Par\'e that the powerful image of any accessible functor F is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption…

This thesis analyses extenders in fine structural mice. Kunen showed that in the inner model for one measurable cardinal, there is a unique measure. This result is generalized, in various ways, to mice below a superstrong cardinal. The…

A left orderable monster is a finitely generated left orderable group all of whose fixpoint-free actions on the line are proximal: the action is semiconjugate to a minimal action so that for every bounded interval $I$ and open interval $J$,…

We describe an obstacle to the analysis of $\mathrm{HOD}^{L[x]}$ as a core model: Assuming sufficient large cardinals, for a Turing cone of reals $x$ there are premice $M,N$ in $\mathrm{HC}^{L[x]}$ such that the pseudo-comparison of $L[M]$…

Let $\kappa$ be a regular cardinal, $\lambda<\kappa$ be a smaller infinite cardinal, and $\mathsf K$ be a $\kappa$-accessible category where colimits of $\lambda$-indexed chains exist. We show that various category-theoretic constructions…

Let $A$ be a finite-dimensional algebra over an algebraically closed field. The problem of constructing indecomposable $A$-modules inductively from simple ones by means of exact sequences - called accessibility - is the starting point of…

We extend A. Miller's framework of $\alpha$-forcing to the case of a regular uncountable cardinal $\kappa = \kappa^{<\kappa}$ and apply it to study the structure of the $\kappa$-Borel hierarchy on subspaces of the generalized Baire space…

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…

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…

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded…