Yasuo Yoshinobu

Fragments of Martin's Maximum in generic extensions

We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly (ω1 + 1)-game-closed forcings. PFA can be destroyed by a strongly (ω1 + 1)-game-closed forcing but not by an ω2-closed. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Directive trees and games on posets

We show that for any infinite cardinal κ, every (κ+1)-strategically closed poset is κ + -strategically closed if and only if □ κ holds. This extends previous results of Velleman, et.al.

Operations, climbability and the proper forcing axiom

Abstract In this paper we show that the Proper Forcing Axiom (PFA) is preserved under forcing over any poset P with the following property: In the generalized Banach–Mazur game over P of length ( ω 1 + 1 ) , Player II has a winning strategy which depends only on the current position and the ordinal indicating the number of moves made so far. By the current position we mean: The move just made by Player I for a successor stage, or the infimum of all the moves made so far for a limit stage. As a consequence of this theorem, we introduce a weak form of the square principle and show that it is consistent with PFA.

Dense non-reflection for stationary collections of countable sets

Abstract We present several forcing posets for adding a non-reflecting stationary subset of P ω 1 ( λ ) , where λ ≥ ω 2 . We prove that PFA is consistent with dense non-reflection in P ω 1 ( λ ) , which means that every stationary subset of P ω 1 ( λ ) contains a stationary subset which does not reflect to any set of size ℵ 1 . If λ is singular with countable cofinality, then dense non-reflection in P ω 1 ( λ ) follows from the existence of squares.

The ⁎-variation of the Banach–Mazur game and forcing axioms

We introduce a property of posets which strengthens (\omega_1+1)-strategic closedness. This property is defined using a variation of the Banach-Mazur game on posets, where the first player chooses a countable set of conditions instead of a single condition at each turn. We prove PFA is preserved under any forcing over a poset with this property. As an application we reproduce a proof of Magidors theorem about the consistency of PFA with some weak variations of the square principles. We also argue how different this property is from (\omega_1+1)-operational closedness, which we introduced in our previous work, by observing which portions of MA^+(\omega_1-closed) are preserved or destroyed under forcing over posets with either property.

σ-short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach-Mazur Boolean game. A σ-short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ-short Boolean algebras and study properties of σ-short Boolean algebras. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)