Related papers: Successive failures of approachability
Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.
Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an $\omega_2$-Aronszajn tree, the $\omega_1$-approachability property fails, and every stationary subset of $\omega_2 \cap \mathrm{cof}(\omega)$…
Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where $\aleph_\omega$ is a strong limit cardinal and the inner model $L(\mathcal{P}(\aleph_\omega))$ satisfies the following properties: (1)…
Starting from the existence of a weakly compact cardinal, we build a generic extension of the universe in which $GCH$ holds and all $\aleph_2$-Aronszajn trees are special and hence there are no $\aleph_2$-Souslin trees. This result answers…
Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $\delta$, $\delta$ is strong limit, $2^\delta=\delta^{+3}$ and the tree property at…
Motivated by the goal of constructing a model in which there are no $\kappa$-Aronszajn trees for any regular $\kappa>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square…
We study the approachability ideal I[\kappa^+] in the context of large cardinals properties of the regular cardinals below a singular \kappa. As a guiding example consider the approachability ideal I[\aleph_{\omega+1}] assuming that…
We solve a long-standing open problem of Shelah regarding the \emph{Approachability Ideal} $I[\kappa^+]$. Given a singular cardinal $\aleph_\gamma$, a regular cardinal $\mu\in (\mathrm{cf}(\gamma),\aleph_\gamma)$ and assuming appropriate…
In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there…
Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal $\delta$ of countable cofinality and every regular uncountable $\mu<\delta$ there are stationarily many non-approachable…
We show that it is consistent from an inaccessible cardinal that classical Namba forcing has the weak $\omega_1$-approximation property. In fact, this is the case if $\aleph_1$-preserving forcings do not add cofinal branches to…
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…
Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal $\mu$, we prove the consistency of the existence of a wide $\mu^+$-Aronszajn tree, i.e. a tree of height and cardinality $\mu^+$ with no branches of…
We prove that every weakly square compact cardinal is a strong limit cardinal. We also study Aronszajn trees with no uncountable finitely branching subtrees, characterizing them in terms of being Lindel\"of with respect to a particular…
Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call ${\rm GM}^+(\omega_3,\omega_1)$ holds. This principle implies ${\rm ISP}(\omega_2)$ and ${\rm ISP}(\omega_3)$, and…
Define the special tree number, denoted $\mathfrak{st}$, to be the least size of a tree of height $\omega_1$ which is neither special nor has a cofinal branch. This cardinal had previously been studied in the context of fragments of…
Assuming the negation of Chang's conjecture, there is a c.c.c. forcing which adds a strongly non-saturated Aronszajn tree. Using a Mahlo cardinal, we construct a model in which there exists a strongly non-saturated Aronszajn tree and the…
We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD,…
We consider a transitive relation on the power set of $\omega_1$ and show if there is a maximal element with respect to this relation then there is a Kurepa tree with no Aronszajn subtree. We also show that if there is a maximal subset of…
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…