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On a conjecture of Lempp

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In this paper, we first prove that there exist computably enumerable (c.e.) degrees a and b such that \({\bf a\not\leq b}\), and for any c.e. degree u, if \({\bf u\leq a}\) and u is cappable, then \({\bf u\leq b}\), so refuting a conjecture of Lempp (in Slaman [1996]); secondly, we prove that: (A. Li and D. Wang) there is no uniform construction to build nonzero cappable degree below a nonzero c.e. degree, that is, there is no computable function \(f\) such that for all \(e\in\omega,\) (i) \(W_{f(e)}\leq_{\rm T}W_e\), (ii) \(W_{f(e)}\) has a cappable degree, and (iii) \(W_{f(e)}\not\leq_{\rm T}\emptyset\) unless \(W_e\leq_{\rm T}\emptyset.\)

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Received: 19 Otober 1998

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Li, A. On a conjecture of Lempp. Arch Math Logic 39, 281–309 (2000).

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  • Computable Function
  • Uniform Construction
  • Cappable Degree