Kleene's O

In set theory and computability theory, Kleene's ${\displaystyle {\mathcal {O}}}$ is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every recursive ordinal, that is, ordinals below Church–Kleene ordinal, ${\displaystyle \omega _{1}^{CK}}$. Since ${\displaystyle \omega _{1}^{CK}}$ is the first ordinal not representable in a computable system of ordinal notations the elements of ${\displaystyle {\mathcal {O}}}$ can be regarded as the canonical ordinal notations.

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