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  1. On Interpretations of Arithmetic and Set Theory.Richard Kaye & Tin Lok Wong - 2007 - Notre Dame Journal of Formal Logic 48 (4):497-510.
    This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies the (...)
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  • The role of true finiteness in the admissible recursively enumerable degrees.Noam Greenberg - 2005 - Bulletin of Symbolic Logic 11 (3):398-410.
    We show, however, that this is not always the case.
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  • The theory of the metarecursively enumerable degrees.Noam Greenberg, Richard A. Shore & Theodore A. Slaman - 2006 - Journal of Mathematical Logic 6 (1):49-68.
    Sacks [23] asks if the metarecursively enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as [Formula: see text] or, equivalently, that of the truth set of [Formula: see text].
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  • Proof theory for theories of ordinals—I: recursively Mahlo ordinals.Toshiyasu Arai - 2003 - Annals of Pure and Applied Logic 122 (1-3):1-85.
    This paper deals with a proof theory for a theory T22 of recursively Mahlo ordinals in the form of Π2-reflecting on Π2-reflecting ordinals using a subsystem Od of the system O of ordinal diagrams in Arai 353). This paper is the first published one in which a proof-theoretic analysis à la Gentzen–Takeuti of recursively large ordinals is expounded.
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