Definability over $\mathrm{B}\hat{I}{\pounds }_2^0$-models

Abstract

Let $\mathfrak{M}=(M,\mathcal{X})$ be a model of $\mathsf{RCA}_0+\text{$Σ^0_2$-bounding}$ in which $\hat{I}{\pounds }_2^0(A)$-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 $\hat{I}{”}_1^0(A)$-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 $A^{\prime }$.

14 pages