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arXiv · 2110.01270

Completeness of the primitive recursive $ω$-rule

Emanuele Frittaion
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Abstract

Shoenfield's completeness theorem (1959) states that every true first order arithmetical sentence has a recursive $ω$-proof encodable by using recursive applications of the $ω$-rule. For a suitable encoding of Gentzen style $ω$-proofs, we show that Shoenfield's completeness theorem applies to cut free $ω$-proofs encodable by using primitive recursive applications of the $ω$-rule. We also show that the set of codes of $ω$-proofs, whether it is based on recursive or primitive recursive applications of the $ω$-rule, is $Π^1_1$ complete. The same $Π^1_1$ completeness results apply to codes of cut free $ω$-proofs.

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BibTeXRIS

Emanuele Frittaion. 2021-10-04. Completeness of the primitive recursive $ω$-rule. https://doi.org/10.1007/s00153-020-00716-9

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