---
tag: WHAP0K600D
kind: concept
name: Kleene's O
sources:
  - wikipedia: en:16105044@r1316619015
    locator: Kleene's O
    note: adopted
    mention: wikipedia://en/16105044
    sections: [Summary]
same_as:
  wikidata: Q6420083
  wikipedia: Kleene's O
inventory: [V644JBE7MV]
---

## Summary

In set theory and computability theory, **Kleene's $\mathcal{O}$** is a canonical subset of the [[natural numbers|0MWKJ44V5A]] 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.