Related papers: A general theory of iterated forcing using finitel…
Based on works of Saharon Shelah, Jakob Kellner, and Anda T\u{a}nasie for controlling the cardinal characteristics of the continuum in ccc forcing extensions, in the author's master's thesis was introduced a new combinatorial notion: the…
The notion of $\theta$-FAM-linkedness, introduced in the second author's master thesis, is a formalization of the notion of strong FAM limits for intervals, whose initial form and applications have appeared in the work of Saharon Shelah,…
Cicho\'n's diagram describes the connections between combinatorial notions related to measure, category, and compactness of sets of irrational numbers. In the second part of the 2010's, Goldstern, Kellner and Shelah constructed a forcing…
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…
We bring forward a logical system of transition algebras that enhances many-sorted first-order logic using features from dynamic logics. The sentences we consider include compositions, unions, and transitive closures of transition…
Ramsey theory and forcing have a symbiotic relationship. At the RIMS Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave three tutorials on Ramsey theory in forcing. The first two tutorials concentrated on…
We develop iterated forcing constructions dual to finite support iterations in the sense that they add random reals instead of Cohen reals in limit steps. In view of useful applications we focus in particular on two-dimensional "random"…
We lay the ground for an Isabelle/ZF formalization of Cohen's technique of forcing. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize the definition of forcing notions as…
The main result of this paper is a partial answer to [math.LO/9909115, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give…
We mechanize, in the proof assistant Isabelle, a proof of the axiom-scheme of Separation in generic extensions of models of set theory by using the fundamental theorems of forcing. We also formalize the satisfaction of the axioms of…
We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper,…
These notes present a compact and self-contained approach to iterated forcing with a particular emphasis on semiproper forcing. We tried to make our presentation accessible to any scholar who has some familiarity with forcing and boolean…
The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories…
The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a…
We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings…
We use coherent systems of FS iterations on a power set, which can be seen as matrix iteration that allows restriction on arbitrary subsets of the vertical component, to prove general theorems about preservation of certain type of unbounded…
We present a detailed general framework to describe the forcing $\tilde{\mathbb{E}}$, defined by Kellner, Shelah and Tan\u{a}sie to prove the consistency with ZFC of an alternative order of Cicho\'n's maximum. Our presentation is close to…
A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is…
In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation. This result shows that for a family of limsup sets,…
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…