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We use the core model for sequences of measures to prove a new lower bound for the consistency strength of the failure of the SCH: THEOREM (i) If there is a singular strong limit cardinal $\kappa$ such that $2^\kappa > kappa^+$ then there…

Under the assumption that $\delta$ is a Woodin cardinal and $\GCH$ holds, I show that if $F$ is any class function from the regular cardinals to the cardinals such that (1) $\kappa<\cf(F(\kappa))$, (2) $\kappa<\lambda$ implies…

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…

For $n<\omega$, we say that the $\Pi^1_n$-reflection principle holds at $\kappa$ and write $\text{Refl}_n(\kappa)$ if and only if $\kappa$ is a $\Pi^1_n$-indescribable cardinal and every $\Pi^1_n$-indescribable subset of $\kappa$ has 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…

Let $\kappa$ be an uncountable cardinal such that $2^{<\kappa} = \kappa$ or just ${\rm cf}(\kappa) > \omega$, $2^{2^{<\kappa}}= 2^\kappa$, and $([\kappa]^\kappa, \supseteq)$ collapses $2^\kappa$ to $\omega$. We show under these assumptions…

We show it is consistent that there is a Souslin tree $S$ such that after forcing with $S$, $S$ is Kurepa and for all clubs $C \subset \omega_1$, $S\upharpoonright C$ is rigid. This answers Fuchs's questions in Club degrees of rigidity and…

Suppose that kappa is a singular cardinal of cofinality omega and GCH holds. Assume that for every n<omega the set of alphas with o(alpha)>= alpha^{+n} is unbounded in kappa.Then there is a cardinal preserving extension satisfying…

We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid…

We investigate the provability of classical combinatorial theorems in ZF. Using combinatorial arguments, we establish the following results for each infinite cardinal ${\kappa}\in On$, (1) ${\kappa}^+\to ({\kappa},{\omega}+1)$, (2) any…

Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the…

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…

Foreman proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of $\omega_1$ is preserved by any proper forcing. We…

We introduce and study a multiplicative analogue of additive indecomposability for linear order types that we call untranscendability, as well as a strengthening that we call $s$-untranscendability. We show that, with the unique exception…

Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal $\aleph_n$, $1 < n <\omega$, is consistent with an arbitrary continuum function below $\aleph_\omega$ which satisfies $2^{\aleph_n} >…

Consider $(\kappa^{+++},\kappa^{++}) \twoheadrightarrow (\kappa^+,\kappa)$ where $\kappa$ is an uncountable regular cardinal. By a result of Shelah's we have $\operatorname{cof}(X \cap \kappa^{++}) = \kappa$ for almost all $X \subset…

For an inaccessible cardinal $\kappa$, the super tree property (ITP) at $\kappa$ holds if and only if $\kappa$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor…

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $\kappa$, updating some classical constructions in the process. This includes models of…

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally…

It is well-known that the consistency strength of the GCH failing at a measurable cardinal is the existence of a cardinal $\kappa$ with $o(\kappa)=\kappa^{++}$. As the literature does not contain more than a proof sketch of the lower bound…