Cross-cuts in the power set of an infinite set

摘要

In the power setP(E)of a setE, the sets of a fixed finite cardinalitykform across-cut, that is, a maximal unordered setCsuch that ifX, Y ⊑EsatisfyX⊑Y, X ⊑someX′inC, andY⊑ someY′inC, thenX⊑Z⊑Yfor someZinC. ForE=ω, ω1, and ω2, it is shown with the aid of the continuum hypothesis thatP(E)has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are p