Completeness of the primitive recursive ω-rule

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 Pi(1)(1) complete. The same Pi(1)(1) completeness results apply to codes of cut free omega-proofs.