More on setwise climbability properties
Bernhard König, Yasuo Yoshinobu
TL;DR
The paper investigates two families of variations on setwise climbability, a fragmentation of Jensen's square principles, by recasting them as Martin-type axioms tied to game-closure properties. The authors distinguish a 'full' family ($SCL^-_f$, $SCL_f$) that aligns with existing principles and remains compatible with $PFA$, from an 'end-extension' family ($SCL^-_e$, $SCL_e$) that corresponds to a $**$-variation of generalized Banach-Mazur games and can fail $PFA$, while still supporting large fragments of forcing axioms. They precisely characterize the end-extension variants as $MA_{\omega_2}$ statements for classes of posets with $**$-closure properties, prove the equivalence of $SCL^-_e$ and $SCL_e$, and explore the interplay with indestructible/absolute properness, including mappings to $AP_{\omega_1}$ and the Mapping Reflection Principle. The results illuminate a nuanced spectrum where certain game-closure forcings preserve core axioms while others destabilize them, and they pose several open questions about implications between these principles and forcing axioms. This work advances understanding of how refined square fragments interact with forcing notions and their preservation properties.
Abstract
We introduce two types of variations of setwise climbability properties, which have been introduced by the second author as fragments of Jensen's square principles. We show that variations of the first type are equivalent to known principles and that they are consistent with the Proper Forcing Axiom(PFA). On the other hand, those of the second type can be characterized as Martin-type axioms for some classes of posets defined in terms of a new variation of generalized Banach-Mazur games, and they are no longer consistent with PFA. We also study how large fragments of PFA are consistent with these principles.