Independence and Induction in Reverse Mathematics

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.

💡 Research Summary

This paper continues the program of analyzing reverse‑mathematical principles that arise from cardinal invariants, focusing on families of objects that are “mutually independent” in a precise sense. Working over the base system RCA₀, the authors investigate four main principles:

1. MAD (Maximal Almost Disjoint) – the existence of a maximal family of pairwise almost‑disjoint subsets of ω.
2. MED (Maximal Eventually Different) – the existence of a maximal family of functions that are pairwise eventually different.
3. DOM (Domination) – for every weakly represented family of total functions there is a single function that eventually dominates every member of the family.
4. HI and BI (High‑immune and bi‑immune Turing degrees) – the existence of a Turing degree that is immune to Σ₂⁰ (high‑immune) or Σ₁⁰ (bi‑immune) sets.

A central technical notion is that of a weakly represented family: a Σ₀¹ partial function Ψ with parameters defines rows Ψₑ, each of which may be a total function; the collection of all total rows forms the family. “Finite” families are those indexed by an M‑finite set (M being the first‑order part of the model), while “non‑finite” families may fail to have the usual infinitary properties when induction is weak.

The authors first formalise DOM and prove a version of Martin’s theorem: over RCA₀, DOM is equivalent to the “high” principle (every Σ₂⁰ set is Δ₂⁰ relative to some B) and to a uniform version where for each A∈S there is a B making every Σ₂⁰(A) set Δ₂⁰(B). Consequently, DOM + BΣ₂⁰ yields full arithmetical induction (IΣₙ⁰ for all n) and therefore PA. However, DOM alone does not imply BΣ₂⁰; a Π₁¹‑conservation result shows RCA₀ + DOM is Π₁¹‑conservative over RCA₀.

The paper then explores the relationship between MAD and DOM. Over BΣ₂⁰, the negation of MAD (¬MAD) is shown to be equivalent to DOM. Intuitively, if there is no maximal almost‑disjoint family, then any weakly represented family of sets can be dominated by a single function, and conversely, a dominating function prevents the construction of a maximal almost‑disjoint family. This establishes a delicate interplay: stronger induction (BΣ₂⁰) makes the existence of a maximal almost‑disjoint family tightly linked to domination.

Turning to functions, the authors introduce MED and the related principles AVOID and ED. AVOID asserts that for every weakly represented family there exists a function eventually different from every member; ED provides such a function explicitly. Under IΣ₂⁰, MED and AVOID turn out to be equivalent, while in weaker systems they separate, illustrating how the strength of induction controls the ability to build maximal eventually‑different families.

Sections 5–7 develop the high‑immune (HI) and bi‑immune (BI) principles. HI states that there is a Turing degree whose every Σ₂⁰ set is avoided (high‑immune); BI strengthens this to avoidance of all Σ₁⁰ sets. The authors show HI is stronger than DOM but still does not reach the strength of ACA₀, and that BI can be obtained via a low bi‑immune degree, leading to a conservation result similar to the one for DOM.

A substantial technical portion (Section 4) presents a “0′′‑tree” injury construction that, given a weakly represented family F and a Π₂⁰ index set B, produces a new weakly represented family G = {Ψₑ : e∈B}. Three lemmas handle Π₀², Σ₀², and Σ₁² index sets respectively, showing how to “scramble” rows of Ψ while preserving totality exactly on the desired indices. This machinery is essential for the later constructions of maximal families under various induction hypotheses.

Finally, the paper assembles these results into a comprehensive picture: the induction strength (IΣ₂⁰, BΣ₂⁰) acts as a crucial parameter governing the existence and equivalence of independence principles. Over weak induction, many of the maximal families fail to exist, while adding BΣ₂⁰ collapses several principles together (¬MAD ↔ DOM, MED ↔ AVOID). The work thus enriches the reverse‑mathematical classification of cardinal‑invariant‑inspired statements, opening avenues for further study of higher‑level induction schemes and other combinatorial invariants.