Completeness of the primitive recursive $$omega $$-rule

Archive for Mathematical Logic:1-17 (forthcoming)

Abstract
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 results apply to codes of cut free \-proofs.
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DOI 10.1007/s00153-020-00716-9
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