Halfway New Cardinal Characteristics

Cardinal characteristics and projective wellorders

Using countable support iterations of S -proper posets, we show that the existence of a ∆3 definable wellorder of the reals is consistent with each of the following: d < c, b < a = s, b < g.

Reductions between Cardinal Characteristics of the Continuum

We discuss two general aspects of the theory of cardinal characteristics of the continuum, especially of proofs of inequalities between such characteristics. The first aspect is to express the essential content of these proofs in a way that makes sense even in models where the inequalities hold trivially (e.g., because the continuum hypothesis holds). For this purpose, we use a Borel version of...

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...

Halfway Houses

Ultrafilters and Cardinal Characteristics of the Continuum

Convention 1. As is customary in set theory, we identify each natural number with the set of its predecessors; in particular, 2 = {0, 1}. We also follow the set-theoretic convention that the set of all natural numbers is denoted by ω. For any sets X and Y , we write X for the set of all functions from Y into X. This convention creates an ambiguity when the sets X and Y are numbers, but the cont...