Definability over $\mathrm BΣ^0_2$-models

TL;DR

This work investigates the definability of solutions to combinatorial problems within weak second-order arithmetic models, focusing on $\mathsf{RCA}_0+\mathrm{B}\Sigma^0_2$ where $\mathrm{I}\Sigma^0_2$ fails. It develops a framework around $\Sigma^0_3(A)$-definable and $\mathrm{B}\Sigma^0_2$-sets, showing these sets are low relative to $A$ (i.e., $G'\le_T A'$) and, in turn, constrains the definability of solutions for Ramsey-type principles. The paper then constructs a $\mathsf{WKL}_0^*$ extension with restricted second-order definability to handle $n\ge 2$ uniformly and derives notable applications: there exist $\Delta^0_1(A)$-instances of $\mathsf{COH}$, $\mathsf{RT}^2_2$, and $\mathsf{TT}^1$ with no arithmetically definable solutions relative to $A$, and every arithmetically definable minimal-degree set relative to $A$ has the same jump as $A'$ (hence is low relative to $A'$). The results illuminate how definability constraints interact with Turing degrees and induce model-theoretic tools for controlling second-order definability in weak bases, offering equivalences linking $\mathrm{I}\Sigma^0_2$ to definable solvability properties and prompting several open questions about stable Ramsey theory and minimal degrees in these contexts.

Abstract

Let $\mathfrak M=(M,\mathcal X)$ be a model of $\mathsf{RCA}_0+\text{$Σ^0_2$-bounding}$ in which $Σ^0_2(A)$-induction fails for some $A\in\mathcal X$. We show that (i) if $\mathfrak M$ is a model of the combinatorial principle Ramsey's Theorem for Pairs, the Cohesive Set Theorem or the Tree Theorem, then there is a $Δ^0_1(A)$-instance of the principle with no solution in $\mathfrak M$ that is arithmetically definable relative to $A$; and (ii) any set of minimal Turing degree in $\mathfrak M$ that is arithmetically definable relative to $A$ has Turing jump equivalent to $A'$.

Theorems & Definitions (21)

• Proposition 2.1: CKWY
• Definition 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Lemma 2.6
• proof
• Theorem 3.1
• proof