Applications of homological algebra to questions in set theory: Gaps and trees.
Daniel Eric Talayco · Deep Blue (University of Michigan) · 1993
Hausdorff gaps and Aronszajn trees are examples of objects which have been very important for set theoretic investigations. This dissertation shows how these objects are in fact homological in nature, developing cohomologies that capture the essential properties of each. For gaps, this leads to questions about the existence of many simultaneous gaps which are answered with both ZFC results and independence results involving new combinatorial hypotheses. For trees, the characterization leads to a new class of Aronszajn trees called Todorcevic trees. A construction demonstrates a connection between these trees and Hausdorff gaps, confirming circumstantial evidence of similarities between these structures. The development of these cohomologies is undertaken with enough generality to be applicable to other set theoretic objects. Other results include: the development and demonstrated consistency of combinatorial hypotheses similar to club which involve stationary sets; a new proof of Hechler's result on dominating subsets of reals which does not use the ramified hierarchy; an application of Hechler's theorem relating the order type of the ideal of non-stationary sets to an unbounded family of reals; a characterization of Souslin trees in terms of a pruning property.