Related papers: Controlling cardinal characteristics without addin…

Much recent work in cardinal characteristics has focused on generalizing results about $\omega$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $\kappa^\kappa$…

Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $\lambda>\kappa$ to $\kappa$) such that the composition still…

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 give a survey of cardinal charcteristics of the higher Cicho\'n diagram defined on the higher Baire space ${}^\kappa\kappa$ for $\kappa$ regular with $2^{<\kappa}=\kappa$. Specifically, we will compare consistency proofs from the…

We will give an overview of four families of cardinal characteristics defined on subspaces $\prod_{\alpha\in\kappa}b(\alpha)$ of the generalised Baire space ${}^\kappa\kappa$, where $\kappa$ is strongly inaccessible and…

Cicho\'n's diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and $\sigma$-compact subsets of the irrationals. It is consistent…

A ccc-generically supercompact cardinal $\kappa$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $\kappa$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically…

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…

We show that from a supercompact cardinal \kappa, there is a forcing extension V[G] that has a symmetric inner model N in which ZF + not AC holds, \kappa\ and \kappa^+ are both singular, and the continuum function at \kappa\ can be…

We show how to construct, via forcing, splitting families than are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different,…

We discuss the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum (like the unbounding or the dominating number or the cardinals related to measure and category on…

We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega^\omega \to \omega^\omega$ introduced by the second author. We prove that while the bounding numbers for these cardinals can be strictly…

Let $\kappa$ be an infinite cardinal. Then, forcing with $\mathbb{R}(\kappa)$$\times$$\mathbb{R}(\kappa)$ adds a generic filter for $\mathbb{C}(\kappa);$ where $\mathbb{R}(\kappa)$ and $\mathbb{C}(\kappa)$ are the forcing notions for adding…

It is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) < \mathrm{add}(\mathrm{Meager})= \mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) = 2^{\aleph_0}. \] Assuming four…

For $\kappa$ regular and uncountable we define variants of the classical cardinal characteristics modulo the non-stationary ideal.

Cardinal characteristics of the continuum represent the boundaries in size between the countable and the continuum with respect to certain properties of sets. They are often defined as the minimum sizes of families of reals that meet some…

Assuming four strongly compact cardinals, it is consistent that all entries in Cicho\'n's diagram are pairwise different, more specifically that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} <…

For a strongly inacessible cardinal $\kappa$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}\kappa$-box product topology - the ideal of "null" sets in the sense of [Sh:1004]…

This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cicho\'n diagram. First I…

For a regular uncountable cardinal kappa, we discuss the order relationship between the unbounding and dominating numbers on kappa and cardinal invariants of the higher meager ideal M_kappa. In particular, we obtain a complete…