Definability over $\mathrm B\Sigma^0_2$-models

TL;DR

This paper investigates the definability and computational properties of models of second-order arithmetic, showing limitations of definability and the behavior of Turing degrees in models where certain induction schemes fail.

Contribution

It demonstrates that in models of RCA_0 with Sigma^0_2-bounding where Sigma^0_2-induction fails, certain combinatorial principles have arithmetically definable instances without solutions, and minimal Turing degrees have jumps equivalent to A'.

Findings

Existence of arithmetically definable instances without solutions.

Minimal Turing degrees in such models have jumps equivalent to A'.

Certain combinatorial principles lack solutions in these models.

Abstract

Let $\mathfrak M=(M,\mathcal X)$ be a model of $\mathsf{RCA}_0+\text{$\Sigma^0_2$-bounding}$ in which $\Sigma^0_2(A)$-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 $\Delta^0_1(A)$-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 $A'$.