General Walks and Their Characteristics

Abstract

One of the most basic questions frequently asked about set-theoretical trees is the question whether they contain any cofinal branch, a branch that intersects each level of the tree. The fundamental importance of this question has already been realized in the work of Kurepa [65] and then later in the works of Erdős and Tarski [32] in their respective attempts to develop the theory of partition calculus and large cardinals. A tree T of height equal to some regular cardinal θ may not have a cofinal branch for a very special reason as the following definition indicates.