Related papers: Diamonds, Compactness, and Measure Sequences
We introduce a new weak variation of diamond that is meant to only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by proving it is compatible with Martin's axiom. We then prove…
This paper makes significant progress towards resolving a conjecture relating strong forcing axioms like $PFA$ and the derived model at a limit of Woodin cardinals $\kappa$. In particular, using a concept called Covering Matrices, we show…
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 present new, streamlined proofs of certain maximality principles studied by Hamkins and Woodin. Moreover, we formulate an intermediate maximality principle, which is shown here to be equiconsistent with the existence of a weakly compact…
We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from…
We prove that the class of all ordinals Ord is not weakly compact with respect to definable classes. Specifically, in any model of ZFC, the definable tree property fails for Ord, in that there is a definable Ord tree with no definable…
We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities of families. The most general form of this compactness notion depends on two ordinal parameters. In the…
A new large-cardinal property is introduced which enables one to give a relative consistency proof of restricted versions of the reflection principles discussed by Tait in his essay "Constructing Cardinals from Below".
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should…
Assume $\kappa = \kappa^{< \kappa}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{\kappa>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this,…
In the first part of the paper, we show that if $\omega \le \kappa < \lambda$ are cardinals, $\kappa^{<\kappa} = \kappa$, and $\lambda$ is weakly compact, then in $V[\M(\kappa,\lambda)]$ the tree property at $\lambda =…
We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. When R is hereditary and of cardinality at most $2^\omega$, we prove that the Weak Diamond and CH…
We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal…
We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals $\kappa \leq \lambda$, where $\kappa$ is the successor of a regular cardinal, there is a rigid saturated ideal on…
We study closure properties of measurable ultrapowers with respect to Hamkin's notion of "freshness" and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one…
Given a Woodin cardinal $\delta$, I show that if $F$ is any Easton function with $F"\delta\subseteq\delta$ and $\GCH$ holds, then there is a cofinality-preserving forcing extension in which $2^\gamma= F(\gamma)$ for each regular cardinal…
Our main goal is here to make a comparative analysis between the well-known MOND theory and a more recent model called$\kappa$-model. An additional connection, between the $\kappa$-model and twoother novel MOND-type theories: Newtonian…
We deal with (< kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues…
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…
We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma$-Prikry. We show that given…