Another Jump Inversion Theorem for Structures

Abstract

In this paper we investigate the question of existence of a jump inversion structure for a given structure $\mathcal{A}$ in the context of their respective degree spectra and the sets definable in them by computable infinitary formulae. More specifically, for a countable structure $\mathcal{A}$ and a computable successor ordinal α, we show that we can apply the construction from [4] to build a structure $\mathcal{N}_\alpha$ such that the sets definable in $\mathcal{A}$ by $\Sigma^{c,\Delta^0_\alpha}_1$ formulae are exactly the sets definable in $\mathcal{N}_\alpha$ by $\Sigma^{c}_{\alpha}$ formulae.