Related papers: Towards Martin's Minimum

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 introduce an iteration of forcing notions satisfying the countable chain condition with minimal damage to a strong coloring. Applying this method, we prove that Martin's axiom is strictly stronger than its restriction to forcing notions…

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $MM^{++}$ of Martin's Maximum does not…

We define a nontrivial version of the square principle $\Box_\omega$, which we then show consistent by means of forcing with finite conditions. This paper has been withdrawn by the author due to the fact that the presented $\Box_\omega$ can…

I show that it is consistent relative to the consistency of a Mahlo cardinal that Martin's axiom holds at $\omega_2$, but the weak Kurepa Hypothesis fails. This answers a question posed by Honzik, Lambie-Hanson and Stejskalov\'a. The…

We give an example of iteration of length omega of (<kappa)-complete kappa^+-cc forcing notions with the limit collapsing kappa^+. The construction is decoded from the proof of Shelah [Proper and Improper Forcing, Appendix, Theorem 3.6(1)].

We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD,…

We study the spectrum of forcing notions between the iterations of $\sigma$-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of $\alpha$-proper forcings for indecomposable countable ordinals as well as…

We introduce a new method for building models of CH, together with $\Pi_2$ statements over $H(\omega_2)$, by forcing. Unlike other forcing constructions in the literature, our construction adds new reals, although only $\aleph_1$-many of…

We investigate properties of trees of height $\omega_1$ and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an $\omega_1$-tree. We introduce fragments of subcompleteness which are…

We show that the weakest versions of Foreman's minimal generic hugeness axioms cannot hold simultaneously on adjacent cardinals. Moreover, conventional forcing techniques cannot produce a model of one of these axioms.

We study L\"owenheim-Skolem and Omitting Types theorems in Transition Algebra, a logical system obtained by enhancing many sorted first-order logic with features from dynamic logic. The sentences we consider include compositions, unions,…

We give some sufficient and necessary conditions on a forcing notion Q for preserving the forcing notion ([omega]^{aleph_0},supseteq^*) is proper. They cover many reasonable forcing notions.

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add…

In this short note, we shall prove some observations regarding the connection between indestructible $\omega_1$-guessing models and the $\omega_1$-approximation property of forcing notions.

In this article we adapt the existing account of class-forcing over a ZFC model to a model $(M,\mathcal{C})$ of Morse-Kelley class theory. We give a rigorous definition of class-forcing in such a model and show that the Definability Lemma…

We introduce a variant of Martin's axiom, called the grounded Martin's axiom, which asserts that the universe is a ccc forcing extension in which Martin's axiom holds for posets in the ground model. This principle already implies several of…

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…

Answering a question of Harrington, we show that there exists a proper forcing notion, which adds a minimal real $\eta \in \prod_{i<\omega} n^*_i$, which is eventually different from any old real in $\prod_{i<\omega} n^*_i$, where the…

For certain weak versions of the Axiom of Choice (most notably, the Boolean Prime Ideal theorem), we obtain equivalent formulations in terms of partial orders, and filter-like objects within them intersecting certain dense sets or…