Honest elementary degrees and degrees of relative provability without the cupping property

Annals of Pure and Applied Logic 168 (5):1017-1031 (2017)
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10.1016/j.apal.2016.11.005

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Provability algebras and proof-theoretic ordinals, I.Lev D. Beklemishev - 2004 - Annals of Pure and Applied Logic 128 (1-3):103-123.
On the induction schema for decidable predicates.Lev D. Beklemishev - 2003 - Journal of Symbolic Logic 68 (1):17-34.

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