Related papers: Strongly compct diagonal Prikry forcing
In this paper we investigate some properties of forcing which can be considered "nice" in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular…
We give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor's identity crisis theorem for the first strongly compact cardinal.
We study which $\kappa$-distributive forcing notions of size $\kappa$ can be embedded into tree Prikry forcing notions with $\kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the…
We prove that Galvin's property consistently fails at successors of strong limit singular cardinals. We also prove the consistency of this property failing at every successor of a singular cardinal. In addition, the paper analyzes the…
We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly…
Let $\mathcal{SN}$ be the $\sigma$-ideal of the strong measure zero sets of reals. We present general properties of forcing notions that allow to control of the additivity of $\mathcal{SN}$ after finite support iterations. This is applied…
We prove that the strong polarized relation for the continuum holds for $\aleph_0$ and for every supercompact cardinal. We use iteration of Mathias forcing.
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…
A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated to arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the…
We consider here Easton support iterations of Prikry type forcing notions. New ways of constructing normal ultrafilters in extensions are presented. It turns out that, in contrast with other supports, seemingly unrelated measures or…
We give an application of our extender based Radin forcing to cardinal arithmetic. Using a preparation forcing and interleaving of Cohen and Levy forcings in the normal Radin sequence we get a model with a power function having a fixed…
This paper presents the main results in my Ph.D. thesis. In what follows several proofs of SCH are presented introducing a family of covering properties which implies both SCH and the failure of various forms of square. These covering…
Using GCH, we force the following: There are continuum many simple cardinal characteristics with pairwise different values.
We present a modification to the Prikry on Extenders forcing notion allowing the blow up of the power set of a large cardinal, change its cofinality to omega without adding bounded subsets, working directly from arbitrary extender (e.g.,…
We continue Gitik, Kanovei and Koepke's work and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.
We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal.…
We try to control many cardinal characteristics by working with a notion of orthogonality between two families of forcings. We show that b^+<g is consistent
We reimplement the creature forcing construction used by Fischer et al. (arXiv:1402.0367) to separate Cicho\'{n}'s diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our…
We characterize sums of normal ultrafilters after the Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower…
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we…