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On the completeness of some transfinite recursive progressions of axiomatic theories

Jens Erik Fenstad

Journal of Symbolic Logic 33 (1):69-76 (1968)

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Transfinite Progressions: A Second Look At Completeness.Torkel Franzén - 2004 - Bulletin of Symbolic Logic 10 (3):367-389.details §1. Iterated Gödelian extensions of theories. The idea of iterating ad infinitum the operation of extending a theory T by adding as a new axiom a Gödel sentence for T, or equivalently a formalization of “T is consistent”, thus obtaining an infinite sequence of theories, arose naturally when Godel's incompleteness theorem first appeared, and occurs today to many non-specialists when they ponder the theorem. In the logical literature this idea has been thoroughly explored through two main approaches. One is that (...) Download Export citation Bookmark 11 citations
Completeness of the primitive recursive $$omega $$ ω -rule.Emanuele Frittaion - 2020 - Archive for Mathematical Logic 59 (5-6):715-731.details Shoenfield’s completeness theorem states that every true first order arithmetical sentence has a recursive \-proof encodable by using recursive applications of the \-rule. For a suitable encoding of Gentzen style \-proofs, we show that Shoenfield’s completeness theorem applies to cut free \-proofs encodable by using primitive recursive applications of the \-rule. We also show that the set of codes of \-proofs, whether it is based on recursive or primitive recursive applications of the \-rule, is \ complete. The same \ completeness (...) Download Export citation Bookmark

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