@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.