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.

Kleene (1938) described a system of notation for all recursive 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 which represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's ${\displaystyle {\mathcal {O}}}$; and given any notation for an ordinal, there is a recursively enumerable set of notations which contains one element for each smaller ordinal and is effectively ordered.

Kleene's ${\displaystyle {\mathcal {O}}}$

The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner. For members ${\displaystyle p}$ of ${\displaystyle {\mathcal {O}}}$, the ordinal for which ${\displaystyle p}$ is a notation is ${\displaystyle |p|}$. The standard definition proceeds via transfinite induction and the ordering ${\displaystyle <_{\mathcal {O}}}$ (a partial ordering of Kleene's ${\displaystyle {\mathcal {O}}}$) is defined simultaneously.

This definition has the advantages that one can recursively enumerate the predecessors of a given ordinal (though not in the ${\displaystyle <_{\mathcal {O}}}$ ordering) and that the notations are downward closed, i.e., if there is a notation for ${\displaystyle \gamma }$ and ${\displaystyle \alpha <\gamma }$ then there is a notation for ${\displaystyle \alpha }$.

Properties of paths in ${\displaystyle {\mathcal {O}}}$

A path in ${\displaystyle {\mathcal {O}}}$ is a subset ${\displaystyle {\mathcal {P}}}$ of ${\displaystyle {\mathcal {O}}}$ which is totally ordered by ${\displaystyle <_{\mathcal {O}}}$ and is closed under predecessors, i.e. if ${\displaystyle p}$ is a member of a path ${\displaystyle {\mathcal {P}}}$ and ${\displaystyle q<_{\mathcal {O}}p}$ then ${\displaystyle q}$ is also a member of ${\displaystyle {\mathcal {P}}}$. A path ${\displaystyle {\mathcal {P}}}$ is maximal if there is no element of ${\displaystyle {\mathcal {O}}}$ which is above (in the sense of ${\displaystyle <_{\mathcal {O}}}$) every member of ${\displaystyle {\mathcal {P}}}$, otherwise ${\displaystyle {\mathcal {P}}}$ is non-maximal.

References

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