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**1 - 2**of**2**### KIMS-2003-07-07 Does Church-Kleene ordinal ωCK1 exist?

, 2003

"... Abstract: A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal ωCK1 really exists. We consider the systems S(α) defined in [2]. Let q̃(α) denote the Gödel number of Rosser formula or its negation A(α) ( = Aq(α)(q (α)) or ¬Aq(α)(q(α))), if the Rosser formula Aq(α)(q( ..."

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Abstract: A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal ωCK1 really exists. We consider the systems S(α) defined in [2]. Let q̃(α) denote the Gödel number of Rosser formula or its negation A(α) ( = Aq(α)(q (α)) or ¬Aq(α)(q(α))), if the Rosser formula Aq(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 < ωCK1, where ω CK 1 is the Church-Kleene ordinal. Lemma. The number q̃(α) is recursively defined for countable recursive ordinals α < ωCK1. Here ‘recursively defined ’ means that q̃(α) is defined inductively starting from 0.

### Does Church-Kleene ordinal ω CK 1 exist?

, 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 ..."

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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.