The Setting

Abstract

Combinatorics with all its various aspects is a broad field of mathematics which has many applications in areas like Topology, Group Theory and even Analysis. A reason for its wide range of applications might be that Combinatorics is rather a way of thinking than a homogeneous theory, and consequently Combinatorics is quite difficult to define. Nevertheless, let us start with a definition of Combinatorics which will be suitable for our purpose:

Combinatorics is the branch of mathematics which studies collections of objects that satisfy certain criteria, and is in particular concerned with deciding how large or how small such collections might be.

Below we give a few examples which should illustrate some aspects of infinitary Combinatorics. At the same time, we present the main topics of this book, which are the Axiom of Choice, Ramsey Theory, cardinal characteristics of the continuum, and forcing.

For one cannot order or compose anything, or understand the nature of the composite, unless one knows first the things that must be ordered or combined, their nature, and their cause.

Gioseffo Zarlino

Le Istitutioni Harmoniche, 1558