A fixed point for the jump operator on structures. (English) Zbl 1305.03036

For a language \(L\), the jump of an \(L\)-structure \(\mathcal A\) is the structure \({\mathcal A}'\) obtained by adding all relations to \(\mathcal A\) that are definable by the computably infinitary \(\Sigma_1\) formulas in the language \(L\). The degree spectrum \(\mathrm{Sp}(\mathcal{A})\) of a structure \(\mathcal{A}\) is the set of Turing degrees that compute a copy of \(\mathcal{A}\). The main result of the paper is the following theorem proved in ZFC+“\(0^{\#}\) exists”:
Theorem 1.4. There is a structure \(\mathcal{A}\) such that \(\mathrm{Sp}({\mathcal A})=\mathrm{Sp}({\mathcal A}')\).
The corresponding structure \(\mathcal A\) is obtained using the Ehrenfeucht-Mostowski theorem. The author notes that Theorem 1.4 was independently proved by V. G. Puzarenko [Algebra Logic 50, No. 5, 418–438 (2011); translation from Algebra Logika 50, No. 5, 615–646 (2011; Zbl 1266.03052)] within ZFC by means of quite different tools. The existence of a structure \(\mathcal A\) in Theorem 1.4 can not be proved in higher-order arithmetic, which is the union of full \(n\)th-order arithmetics for all \(n\) (i.e., the theory of the structure \((\omega,{\mathcal P}(\omega),{\mathcal P}({\mathcal P}(\omega)),\ldots;0,1,+,\times,<,\in)\)). Namely, the following theorem is proved in full second-order arithmetic:
Theorem 1.5. The existence of a structure \(\mathcal A\) with \(\mathrm{Sp}({\mathcal A})=\mathrm{Sp}({\mathcal A}')\) implies the consistency of higher-order arithmetic.


03C57 Computable structure theory, computable model theory
03F35 Second- and higher-order arithmetic and fragments


Zbl 1266.03052
Full Text: DOI arXiv Euclid


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