Stationary Reflection on Pₖ (λ)

Abstract

Throughout history, mathematicians have had to deal with infinity, always considering it in the “potential” sense, rather than an actual object. It was not until the late nineteenth century that actual infinity was the subject matter. In 1874 George Cantor published “On a Property of the Collection of All Real Algebraic Numbers”. From the results he proved in that paper, he concluded that there were larger infinites than others, giving birth in this way to Set Theory, the study of infinite sets and the set-theoretic foundations of mathematics. The study of infinite sets, and in particular their combinatorial properties, is not only of interest in itself, but it has numerous applications in areas such as analysis, algebra and topology (see e.g. [1; 2; 3]). Even possible applications to mathematical biology have being studied [4]. Combinatorics is always concerned about sizes, and when dealing with infinite sets there are different ways to capture the idea of how large a set is. For example, the notion of “filter” on a set A corresponds to “big” subsets of A, while positive subsets in the sense of a given filter corresponds to the notion of “not small”. Stationary subsets of a cardinal k are those that are not small in the sense of the closed and unbounded filter of k.