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

中文翻译：

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基本递归$$ \ omega $$ω-规则的完整性

Shoenfield的完备性定理（1959）指出，每个真实的一阶算术句子都有一个递归\（\ omega \）证明，可通过使用\（\ omega \）规则的递归应用来编码。对于根岑风格合适的编码\（\欧米茄\） -proofs，我们表明，Shoenfield完备性定理适用于切取游离\（\欧米茄\） -proofs地编码使用的原始递归应用的\（\欧米茄\） -rule 。我们还表明，一组代码的\（\欧米茄\） -proofs，无论是基于递归或原始递归应用的\（\欧米茄\） -rule，是\（\ varPi ^ 1_1 \）完成。相同\（\ varPi ^ 1_1 \）完整性结果适用于免剪切\（\ omega \）证明的代码。