Simple Cardinal Characteristics of the Continuum

We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (Σ 2 and, under suitable restrictions, Π 2 ) are shown to have pleasant properties, related to Baire category. We construct models of set theory where (unrestricted) Π 2 -characteristics behave quite chaotically and no new characteristics appear at higher complexity levels. We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified proof of Shelah’s theorem that the dominating number is less than or equal to the independence number.