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Independence and Induction in Reverse Mathematics

David Belanger, Chi Tat Chong, Rupert Hölzl, Frank Stephan

TL;DR

This paper investigates independence-inspired principles in reverse mathematics, focusing on MAD and MED within the $\mathsf{RCA}_0$ framework and the role of induction strength at $\mathsf{B}\Sigma^0_2$ and $\mathsf{I}\Sigma^0_2$. It develops the theory of weakly represented families, introduces $0''$-injury techniques to handle index sets, and establishes a tight connection between domination and maximal independence via $\mathsf{DOM}$ and $\mathsf{MAD}$. It also analyzes avoidance, eventual difference, hyperimmunity, and bi-immunity, providing model-theoretic and conservation results that separate these principles and illuminate the interaction between combinatorial cardinal-invariant notions and logical strength. Overall, the work clarifies how induction strength shapes the existence of large independent families and related degree-theoretic properties in reverse mathematics, with implications for the relative strength of $\mathsf{DOM}$, $\mathsf{MAD}$, $\mathsf{MED}$, and associated principles.

Abstract

We continue the project of the study of reverse mathematics principles inspired by cardinal invariants. In this article in particular we focus on principles encapsulating the existence of large families of objects that are in some sense mutually independent. More precisely, we study the principle $\mathsf{MAD}$ stating that a maximal family of pairwise almost disjoint sets exists; and the principle $\mathsf{MED}$ expressing the existence of a maximal family of functions that are pairwise eventually different. We investigate characterisations of and relations between these principles and some of their variants. It turns out that induction strength at the levels of $\mathsf{B}\mathrmΣ_2^0$ or $\mathsf{I}\mathrmΣ_2^0$ is an essential parameter; for instance, over $\mathsf{B}\mathrmΣ_2^0$, we show that $\neg\mathsf{MAD}$ is equivalent to the principle $\mathsf{DOM}$ expressing that every weakly represented family of functions is dominated by some other function.

Independence and Induction in Reverse Mathematics

TL;DR

This paper investigates independence-inspired principles in reverse mathematics, focusing on MAD and MED within the framework and the role of induction strength at and . It develops the theory of weakly represented families, introduces -injury techniques to handle index sets, and establishes a tight connection between domination and maximal independence via and . It also analyzes avoidance, eventual difference, hyperimmunity, and bi-immunity, providing model-theoretic and conservation results that separate these principles and illuminate the interaction between combinatorial cardinal-invariant notions and logical strength. Overall, the work clarifies how induction strength shapes the existence of large independent families and related degree-theoretic properties in reverse mathematics, with implications for the relative strength of , , , and associated principles.

Abstract

We continue the project of the study of reverse mathematics principles inspired by cardinal invariants. In this article in particular we focus on principles encapsulating the existence of large families of objects that are in some sense mutually independent. More precisely, we study the principle stating that a maximal family of pairwise almost disjoint sets exists; and the principle expressing the existence of a maximal family of functions that are pairwise eventually different. We investigate characterisations of and relations between these principles and some of their variants. It turns out that induction strength at the levels of or is an essential parameter; for instance, over , we show that is equivalent to the principle expressing that every weakly represented family of functions is dominated by some other function.
Paper Structure (8 sections, 20 theorems, 25 equations)

This paper contains 8 sections, 20 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dominating families of functions
  4. Handling index sets
  5. Maximal almost disjoint families of sets
  6. Avoidance and eventual difference
  7. Hyperimmunity in Reverse Mathematics
  8. Bi-immunity

Key Result

Lemma 2

For each $A \in \mathcal{S}$ there exists a $\Sigma^A_1$ partial function $\Phi^A \colon M^2 \rightarrow M$ which is universal in the sense that for every $\Sigma^A_1$ partial function $\Psi^A \colon M^2 \rightarrow M$ there is a $\Delta_1^A$ function $f$ satisfying $\Psi^A_e = \Phi^A_{f(e)}$.

Theorems & Definitions (39)

  • Definition 1: Weakly represented families
  • Lemma 2
  • Definition 3
  • Proposition 5: Martin's Theorem, formalised
  • proof
  • Theorem 6: Hölzl, Raghavan, Stephan, Zhang HRSZ17
  • Lemma 7
  • proof : Proof of Theorem \ref{['dom-induction']}
  • Theorem 8
  • proof
  • ...and 29 more