In set theory and computability theory, Kleene's
is a canonical subset of the natural numbers when regarded as ordinal notations.
is the first ordinal not representable in a computable system of ordinal notations the elements of
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
; 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.
The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner.
(a partial ordering of Kleene's
) are the smallest sets such that the following holds.
[citation needed] 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
There are alternate definitions, such as the set of indices of (partial) well-orderings of the natural numbers.
is called a notation and is meant to give a definition of the ordinal
, a successor ordinal is defined in terms of a notation for the ordinal immediately preceding it.
Specifically, a successor notation
, a notation for a limit ordinal
amounts to a computable sequence of notations for smaller ordinals limiting to
'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
by repeatedly applying the operations
is eventually referenced in the definition of
by repeatedly applying the operations
is computably enumerable: if
, then a computer program will eventually find a proof of this fact.
only when all of the criteria below are met: A path in