In set theory and computability theory, Kleene's O {\displaystyle {\mathcal {O}}} is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every computable ordinal, that is, ordinals below Church–Kleene ordinal, ω 1 CK {\displaystyle \omega _{1}^{\text{CK}}} . Since ω 1 CK {\displaystyle \omega _{1}^{\text{CK}}} is the first ordinal not representable in a computable system of ordinal notations the elements of O {\displaystyle {\mathcal {O}}} can be regarded as the canonical ordinal notations. Kleene (1938) described a system of notation for all computable ordinals (those less than the Church–Kleene ordinal). It uses a subset of the natural numbers instead of finite strings of symbols. Unfortunately, there is in general no effective way to tell whether some natural number represents an ordinal, or whether two numbers represent the same ordinal. However, one can effectively find notations that represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's O {\displaystyle {\mathcal {O}}} ; and given any notation for an ordinal, there is a computably enumerable set of notations that contains one element for each smaller ordinal and is effectively ordered.
Definition The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner. For members p {\displaystyle p} of O {\displaystyle {\mathcal {O}}} , the ordinal for which p {\displaystyle p} is a notation is | p | {\displaystyle |p|} . O {\displaystyle {\mathcal {O}}} and < O {\displaystyle <_{\mathcal {O}}} (a partial ordering of Kleene's O {\displaystyle {\mathcal {O}}} ) are the smallest sets such that the following holds.
0 ∈ O ∧ | 0 | = 0 {\displaystyle 0\in {\mathcal {O}}\land |0|=0} .
i ∈ O ∧ | i | = α → 2 i ∈ O ∧ | 2 i | = α + 1 ∧ i < O 2 i {\displaystyle i\in {\mathcal {O}}\land |i|=\alpha \rightarrow 2^{i}\in {\mathcal {O}}\land |2^{i}|=\alpha +1\land i<_{\mathcal {O}}2^{i}}
Suppose { e } {\displaystyle \{e\}} is the e {\displaystyle e} -th partial computable function. If e {\displaystyle e} is total and range ( { e } ) ⊆ O ∧ ∀ n ( { e } ( n ) < O { e } ( n + 1 ) ) {\displaystyle {\textrm {range}}(\{e\})\subseteq {\mathcal {O}}\land \forall n(\{e\}(n)<_{\mathcal {O}}\{e\}(n+1))} , then 3 ⋅ 5 e ∈ O ∧ ∀ n , { e } ( n ) < O 3 ⋅ 5 e ∧ | 3 ⋅ 5 e | = lim k | { e } ( k ) | {\displaystyle 3\cdot 5^{e}\in {\mathcal {O}}\land \forall n,\{e\}(n)<_{\mathcal {O}}3\cdot 5^{e}\land |3\cdot 5^{e}|=\lim _{k}|\{e\}(k)|}
p < O q ∧ q < O r → p < O r {\displaystyle p<_{\mathcal {O}}q\land q<_{\mathcal {O}}r\rightarrow p<_{\mathcal {O}}r}
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