We give a characterization of the cofinality of the strong measure zero ideal under the continuum hypothesis and prove that we can force it to a value less than the power of the continuum.

We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property . In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber 1 and for forcing notions with the (...) property fails. This negatively answers a part of one of the classical problems about implications between fragments of. (shrink)

Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _2$$\end{document}. The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the Proper Forcing Axiom but implies some important conclusions from the Proper Forcing Axiom, (...) for example, the P-ideal Dichotomy. In this article, it is proved that the Y-Proper Forcing Axiom implies the Mapping Reflection Principle by introducing forcing notions whose conditions are finite objects. (shrink)

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ1 dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ1 dense sets related to the rectangle refining property, and prove that they are really weaker fragments.

In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees , Proposition 2.7 is not true. To avoid this error and correct Proposition 2.7, the definition of the property is changed. In Yorioka [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. We give a new statement and a new proof of Lemma 6.9.

We prove that the covering number of the Marczewski ideal is equal to ℵ1 in the extension with the iteration of Hechler forcing.

In this paper we build an Asperó–Mota iteration of length $\omega _2$ that adds a family of $\aleph _2$ many club subsets of $\omega _1$ which cannot be diagonalized while preserving $\aleph _2$. This result discloses a technical limitation of some types of Asperó–Mota iterations.

One of the main problems about is that whether a coherent Suslin tree forces that there are no S‐spaces under. We analyze a forcing notion related to this problem, and show that under, S forces that every topology on ω1 generated by a basis in the ground model is not an S‐topology. This supplements the previous work due to Stevo Todorčević [25].

We show that, under PFA, a coherent Suslin tree forces that every two Aronszajn trees are club-isomorphic.

We prove the iteration lemmata, which are the key lemmata to show that extensions by Pmax variations satisfy absoluteness for Π2-statements in the structure 〈H , ∈, NSω 1, R 〉 for some set R of reals in L , for the following statements: The cofinality of the null ideal is ℵ1. There exists a good basis of the strong measure zero ideal.

We prove the theorem from the title which answers a question addressed in the paper of Moore-Hrusak-Dzamonja [3].

Supplementing the well known results of Kunen we show that Martin’s Axiom is not sufficient to decide the existence of -gaps when -gaps exist, that is, it is consistent with ZFC that Martin’s Axiom holds and there are -gaps but no -gaps.