In set theory and computability theory, Kleene's 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, . Since 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 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 ; 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.
The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner. For members of , the ordinal for which is a notation is . The standard definition proceeds via transfinite induction and the ordering (a partial ordering of Kleene's ) is defined simultaneously.
- The natural number 0 belongs to Kleene's and .
- If belongs to Kleene's and , then belongs to Kleene's and and .
- Suppose is the -th partial recursive function. If is total, with range contained in , and for every natural number , we have , then belongs to Kleene's , for each and , i.e. is a notation for the limit of the ordinals where for every natural number .
- and imply (this guarantees that is transitive.)
This definition has the advantages that one can recursively 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 and then there is a notation for .
Basic properties of
- If and and then ; but the converse may fail to hold.
- induces a tree structure on , so is well-founded.
- only branches at limit ordinals; and at each notation of a limit ordinal, is infinitely branching.
- Since every recursive function has countably many indices, each infinite ordinal receives countably many notations; the finite ordinals have unique notations, usually denoted .
- The first ordinal that doesn't receive a notation is called the Church–Kleene ordinal and is denoted by . Since there are only countably many recursive functions, the ordinal is evidently countable.
- The ordinals with a notation in Kleene's are exactly the recursive ordinals. (The fact that every recursive ordinal has a notation follows from the closure of this system of ordinal notations under successor and effective limits.)
- is not recursively enumerable, but there is a recursively enumerable relation which agrees with precisely on members of .
- For any notation , the set of notations below is recursively enumerable. However, Kleene's , when taken as a whole, is (see analytical hierarchy).
- In fact, is -complete and every subset of is effectively bounded in (a result of Spector).
- is the universal system of ordinal notations in the sense that any specific set of ordinal notations can be mapped into it in a straightforward way. More precisely, there is a recursive function such that if is an index for a recursive well-ordering, then is a member of and is order-isomorphic to an initial segment of the set .
- There is a recursive function , which, for members of , mimics ordinal addition and has the property that . (Jockusch)
Properties of paths in
A path in is a subset of which is totally ordered by and is closed under predecessors, i.e. if is a member of a path and then is also a member of . A path is maximal if there is no element of which is above (in the sense of ) every member of , otherwise is non-maximal.
- A path is non-maximal if and only if is recursively enumerable (r.e.). It follows by remarks above that every element of determines a non-maximal path ; and every non-maximal path is so determined.
- There are maximal paths through ; since they are maximal, they are non-r.e.
- In fact, there are maximal paths within of length . (Crossley, Schütte)
- For every non-zero ordinal , there are maximal paths within of length . (Aczel)
- Further, if is a path whose length is not a multiple of then is not maximal. (Aczel)
- For each r.e. degree , there is a member of such that the path has many-one degree . In fact, for each recursive ordinal , a notation exists with . (Jockusch)
- There exist paths through which are . Given a progression of recursively enumerable theories based on iterating Uniform Reflection, each such path is incomplete with respect to the set of true sentences. (Feferman & Spector)
- There exist paths through each initial segment of which is not merely r.e., but recursive. (Jockusch)
- Various other paths in have been shown to exist, each with specific kinds of reducibility properties. (See references below)