We show that under ZFC, for every indecomposable ordinal α < ω₁, there exists a poset which is β-proper for every β < α but not α-proper. It is also shown that a poset is forcing equivalent to a poset satisfying Axiom A if and only if it is α-proper for every α < ω₁.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-2,
author = {Tetsuya Ishiu},
title = {$\alpha$-Properness and Axiom A},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {25-37},
zbl = {1079.03039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-2}
}
Tetsuya Ishiu. α-Properness and Axiom A. Fundamenta Mathematicae, Tome 185 (2005) pp. 25-37. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-2/