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α-Properness and Axiom A

100%

Ishiu T.

Fundamenta Mathematicae

2005

tom 186

nr 1

25-37

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.

Minimality of non-σ-scattered orders

63%

Ishiu T., Moore J.

Fundamenta Mathematicae

2009

tom 205

nr 1

29-44

We will characterize-under appropriate axiomatic assumptions-when a linear order is minimal with respect to not being a countable union of scattered suborders. We show that, assuming PFA⁺, the only linear orders which are minimal with respect to not being σ-scattered are either Countryman types or real types. We also outline a plausible approach to demonstrating the relative consistency of: There are no minimal non-σ-scattered linear orders. In the process of establishing these results, we will prove combinatorial characterizations of when a given linear order is σ-scattered and when it contains either a real or Aronszajn type.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Ishiu T.

1 Moore J.

1 2009

1 2005

Wyniki wyszukiwania

α-Properness and Axiom A

Minimality of non-σ-scattered orders