Applications of cohomology to set theory. II: Todorčević trees
The author develops a cohomology theory for a class of \(\omega_1\)-trees. This continues the author's investigations in Part I [ibid. 71, 69-106 (1995; Zbl 0824.03029)] on application of cohomology to gaps. An \(\omega_1\)-tree \(T\) is special if there is a function \(f: T\to \mathbb{Z}\) such that if \(s< t\) then \(f(s)\neq f(t)\). An Aronszajn tree is an \(\omega_1\)-tree with no uncountable chain. It is well known that MA implies that each Aronszajn tree is special. It was observed by Todorčević that special Aronszajn trees can be coded by partitions of pairs of countable ordinals into countably many pieces. This fact is used by the author to apply cohomology to a class of \(\omega_1\)-trees called Todorčević trees. The author shows that there are similarities in the construction of Hausdorff gaps and Todorčević trees. He shows that under \(\text{MA} +\neg \text{CH}\) all Aronszajn trees are special Todorčević trees whereas under \(\diamondsuit\) there are Todorčević trees which are not special Aronszajn trees.
scientific article; zbMATH DE number 908464
scientific article