## 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.

### MSC:

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

Zbl 1266.03052
Full Text:

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