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On recursion theory in \(I\Sigma_ 1\). (English) Zbl 0703.03019
In 1986, the second author showed a priority-free proof of the existence of an intermediate r.e. degree [Lect. Notes Comput. Sci. 233, 493-500 (1986; Zbl 0615.03033)]. In the present paper, the authors discuss the problem of whether Kučera’s priority-free proof formalizes in the fragment of first order arithmetic \(I\Sigma_ 1\) (the basic theory \(PA^-\) plus induction for \(\Sigma_ 1\) formulas). It is shown that the low basis theorem is meaningful and provable in \(I\Sigma_ 1\) and that the priority-free solution to Post’s problem formalizes in this theory.
Reviewer: Li Xiang

MSC:
03D25 Recursively (computably) enumerable sets and degrees
03F30 First-order arithmetic and fragments
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References:
[1] \(\Sigma\) 2-collection and the infinite injury priority method 53 pp 212– (1988)
[2] Finite injury and \(\Sigma\) 1-induction 54 pp 38– (1989) · Zbl 0671.03029
[3] DOI: 10.1007/BFb0016275 · doi:10.1007/BFb0016275
[4] Logic Colloquium ’77 pp 199– (1978)
[5] DOI: 10.1007/BFb0076215 · doi:10.1007/BFb0076215
[6] Methods in mathematical logic (proceedings of the sixth Latin American symposium, Caracas, 1983) 1130 pp 32– (1983)
[7] Transactions of the American Mathematical Society 173 pp 173– (1972) · Zbl 0247.00014
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