$\alpha $-Properness and Axiom A
Volume 186 / 2005
Fundamenta Mathematicae 186 (2005), 25-37
MSC: Primary 03E40; Secondary 03E35.
DOI: 10.4064/fm186-1-2
Abstract
We show that under ZFC, for every indecomposable ordinal $\alpha<\omega_1$, there exists a poset which is $\beta$-proper for every $\beta<\alpha$ but not $\alpha$-proper. It is also shown that a poset is forcing equivalent to a poset satisfying Axiom A if and only if it is $\alpha$-proper for every $\alpha<\omega_1$.
