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The cardinal characteristics of the ideal generated by the $F_σ$ measure zero subsets of the reals

Miguel A. Cardona

Abstract

Let $\mathcal{E}$ be the ideal generated by the $F_σ$ measure zero subsets of the reals. The purpose of this survey paper is to study the cardinal characteristics (the additivity, covering number, uniformity, and cofinality) of $\mathcal{E}$.

The cardinal characteristics of the ideal generated by the $F_σ$ measure zero subsets of the reals

Abstract

Let be the ideal generated by the measure zero subsets of the reals. The purpose of this survey paper is to study the cardinal characteristics (the additivity, covering number, uniformity, and cofinality) of .
Paper Structure (4 sections, 29 theorems, 45 equations, 18 figures)

This paper contains 4 sections, 29 theorems, 45 equations, 18 figures.

Table of Contents

  1. Introduction
  2. The consistency of and
  3. The consistency of
  4. The consistency of

Key Result

Theorem 1.1

Figures (18)

  • Figure 1: Diagram of the cardinal characteristics associated with $\mathcal{I}$. An arrow $\mathfrak x\rightarrow\mathfrak y$ means that (provably in ZFC) $\mathfrak x\le\mathfrak y$.
  • Figure 2: Cichoń's diagram and the cardinal characteristics associated with $\mathcal{E}$.
  • Figure 3: The constellation of Cichoń's diagram after adding $\lambda=\lambda^{\aleph_0}$ many Cohen reals.
  • Figure 4: Cichon's diagram after adding $\kappa$-many eventually different reals with $\mathbb{E}$.
  • Figure 5: Possible models where the four cardinal characteristics associated with $\mathcal{E}$ can be pairwise different.
  • ...and 13 more figures

Theorems & Definitions (57)

  • Theorem 1.1: BS1992, see also BJ
  • Theorem 1.2
  • Definition 1.3
  • Example 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 47 more