Does Church-kleene Ordinal Ω Exist?

A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal ω 1 really exists. We consider the systems S defined in [2]. Let q̃(α) denote the Gödel number of Rosser formula or its negation A(α) (= Aq(α)(q ) or ¬Aq(α)(q )), if the Rosser formula Aq(α)(q ) is well-defined. By “recursive ordinals” we mean those defined by Rogers [4]. Then that α is a recursive ordinal means that α < ω 1 , where ω CK 1 is the ChurchKleene ordinal. Lemma. The number q̃(α) is recursively defined for countable recursive ordinals α < ω 1 . Here ‘recursively defined’ means that q̃(α) is defined inductively starting from 0. Remark. The original meaning of ‘recursive’ is ‘inductive.’ The meaning of the word ‘recursive’ in the following is the one that matches the spirit of Kleene [3] (especially, the spirit of the inductive construction of metamathematical predicates described in section 51 of [3]). Proof. The well-definedness of q̃(0) is assured by Rosser-Gödel theorem as explained in [2]. We make an induction hypothesis that for each δ < α, the Gödel number q̃(γ) of the formula A(γ) (= Aq(γ)(q ) or ¬Aq(γ)(q )) with γ ≤ δ is recursively defined for γ ≤ δ.