@MISC{Kitada03doeschurch-kleene, author = {Hitoshi Kitada}, title = {Does Church-Kleene ordinal ω CK 1 exist?}, year = {2003} }

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Abstract: A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal ω CK 1 really exists. We consider the systems S (α) defined in [2]. Let ˜q(α) denote the Gödel number of Rosser formula or its negation A (α) ( = A q (α)(q (α) ) or ¬A q (α)(q (α))), if the Rosser formula A q (α)(q (α) ) is well-defined. By “recursive ordinals ” we mean those defined by Rogers [4]. Then that α is a recursive ordinal means that α < ω CK 1, where ω CK 1 is the Church-Kleene ordinal.

... number of Rosser formula or its negation A (α) (= A q (α)(q (α) ) or ¬A q (α)(q (α) )), if the Rosser formula A q (α)(q (α) ) is well-defined. By “recursive ordinals” we mean those defined by Rogers =-=[4]-=-. Then that α is a recursive ordinal means that α < ω CK 1 , where ω CK 1 is the ChurchKleene ordinal. Lemma. The number ˜q(α) is recursively defined for countable recursive ordinals α < ω CK 1 . Here...

...) 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 ˜q(0) is assured by Rosser-Gödel theorem as explai...

...il: kitada@ms.u-tokyo.ac.jp July 7, 2003 Abstract: A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal ω CK 1 really exists. We consider the systems S (α) defined in =-=[2]-=-. Let ˜q(α) denote the Gödel number of Rosser formula or its negation A (α) (= A q (α)(q (α) ) or ¬A q (α)(q (α) )), if the Rosser formula A q (α)(q (α) ) is well-defined. By “recursive ordinals” we m...