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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
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