Kleene's O

Summary

In set theory and computability theory, Kleene's $\mathcal{O}$ 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, $\omega_1^{\text{CK}}$. Since $\omega_1^{\text{CK}}$ is the first ordinal not representable in a computable system of ordinal notations the elements of $\mathcal{O}$ can be regarded as the canonical ordinal notations.