ON THE NOTION OF JUMP STRUCTURE

Abstract

For a given countable structure $\mathfrak{A}$ and a computable ordinal $\alpha$, we define its $\alpha$-th jump structure ${\mathfrak{A}}^{(\alpha )}$. We study how the jump structure relates to the original structure. We consider a relation between structures called conservative extension and show that ${\mathfrak{A}}^{(\alpha )}$ conservatively extends the structure $\mathfrak{A}$. It follows that the relations definable in $\mathfrak{A}$ by computable infinitary $\sum _{\alpha }$ formulae are exactly the relations definable in ${\mathfrak{A}}^{(\alpha )}$ by computable infinitary $\sum _1$ formulae. Moreover, the Turing degree spectrum of ${\mathfrak{A}}^{(\alpha )}$ is equal to the $\alpha$′-th jump Turing degree spectrum of $\mathfrak{A}$, where $\alpha \prime =\alpha +1,\text{ if }\alpha <\omega \text{, and }\alpha \prime =\alpha$, otherwise.