Twelve Cardinals and Their Relations

Abstract

In this chapter we investigate twelve cardinal characteristics and their relations to one another. A cardinal characteristic of the continuum is an uncountable cardinal number which is less than or equal to \(\mathfrak {c}\) that describes a combinatorial or analytical property of the continuum. Like the power of the continuum itself, the size of a cardinal characteristic is often independent from ZFC. However, some restrictions on possible sizes follow from ZFC, and we shall give a complete list of what is known to be provable in ZFC about their relation. Later in Part II, but mainly in Part III, we shall see how one can diminish or augment some of these twelve cardinals without changing certain other cardinals. In fact, these cardinal characteristics are also used to investigate combinatorial properties of the various forcing notions introduced in Part III.We shall encounter some of these cardinal characteristics (e.g., \(\mathfrak {p}\)) more often than others (e.g., \(\mathfrak {i}\)). However, we shall encounter each of these twelve cardinals again, and like the twelve notes of the chromatic scale, these twelve cardinals will build the framework of our investigation of the combinatorial properties of forcing notions that is carried out in Part III.KeywordsUncountable Cardinal NumberCardinal CharacteristicsMaximal Independent FamilyArbitrary Infinite CardinalityUnbounded FamilyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.