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,. Since 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. 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 which represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's \mathcal{O}; and given any notation for an ordinal, there is a computably enumerable set of notations which 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 of \mathcal{O}, the ordinal for which p is a notation is | p |. \mathcal{O} and (a partial ordering of Kleene's \mathcal{O}) are the smallest sets such that the following holds. This definition has the advantages that one can computably enumerate the predecessors of a given ordinal (though not in the ordering) and that the notations are downward closed, i.e., if there is a notation for \gamma and then there is a notation for \alpha. There are alternate definitions, such as the set of indices of (partial) well-orderings of the natural numbers.

Explanation

A member p of Kleene's \mathcal{O} is called a notation and is meant to give a definition of the ordinal |p|. The successor notations are those such that |p| is a successor ordinal \alpha + 1. In Kleene's \mathcal{O}, a successor ordinal is defined in terms of a notation for the ordinal immediately preceding it. Specifically, a successor notation p is of the form 2^q for some other notation q, so that. The limit notations are those such that |p| is a limit ordinal. In Kleene's \mathcal{O}, a notation for a limit ordinal \beta amounts to a computable sequence of notations for smaller ordinals limiting to \beta. Any limit notation p is of the form 3\cdot 5^e where the e'th partial computable function is a total function listing an infinite sequence of notations, which are strictly increasing under the order. The limit of the sequence of ordinals is |p|. Although implies |q| < |p|, |q| < |p| does not imply. In order for, q must be reachable from p by repeatedly applying the operations and for. In other words, when q is eventually referenced in the definition of |p| given by p.

A Computably Enumerable Order Extending the Kleene Order

For arbitrary, say that q\prec p when q is reachable from p by repeatedly applying the operations and for. The relation \prec agrees with on \mathcal{O}, but \prec is computably enumerable: if q\prec p, then a computer program will eventually find a proof of this fact. For any notation, all q\prec p are themselves notations in \mathcal{O}. For, p is a notation of \mathcal{O} only when all of the criteria below are met:

Basic properties of <O

Properties of paths in <O

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