# zbMATH — the first resource for mathematics

Jump inversions of algebraic structures and $$\Sigma$$-definability. (English) Zbl 07197304
Summary: It is proved that for every countable structure $$\mathcal{A}$$ and a computable successor ordinal $$\alpha$$ there is a countable structure $$\mathcal{A}^{-\alpha}$$ which is $$\leq _{\Sigma}$$-least among all countable structures $$\mathcal{C}$$ such that $$\mathcal{A}$$ is $$\Sigma$$-definable in the $$\alpha$$th jump $$\mathcal{C}^{(\alpha)}$$. We also show that this result does not hold for the limit ordinal $$\alpha=\omega$$. Moreover, we prove that there is no countable structure $$\mathcal{A}$$ with the degree spectrum $$\{\mathbf{d}:\mathbf{a}\leq\mathbf{d}^{(\omega)}\}$$ for $$\mathbf{a}>\mathbf{0}^{(\omega)}$$.

##### MSC:
 03 Mathematical logic and foundations
Full Text: