Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
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We study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by 驴be, which is a natural extension of s-reducibility 驴s. We show that 驴s, 驴be, and enumeration reducibility do not coincide on the $${\Pi^0_1}$$ ---sets, and the structure $${\boldsymbol{\mathcal{D}_{\rm be}}}$$ of the be-degrees is not elementarily equivalent to the structure of the s-degrees. We show also that the first order theory of $${\boldsymbol{\mathcal{D}_{\rm be}}}$$ is computably isomorphic to true second order arithmetic: this answers an open question raised by Cooper (Z Math Logik Grundlag Math 33:537---560, 1987).