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 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 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.

Definition

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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  .   and   (a partial ordering of Kleene's  ) are the smallest sets such that the following holds.[citation needed]

  •  .
  •  
  • Suppose   is the  -th partial computable function. If   is total and  , then  
  •  

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   and   then there is a notation for  . There are alternate definitions, such as the set of indices of (partial) well-orderings of the natural numbers.[1]

Explanation

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A member   of Kleene's   is called a notation and is meant to give a definition of the ordinal  .

The successor notations are those such that   is a successor ordinal  . In Kleene's  , a successor ordinal is defined in terms of a notation for the ordinal immediately preceding it. Specifically, a successor notation   is of the form   for some other notation  , so that  .

The limit notations are those such that   is a limit ordinal. In Kleene's  , a notation for a limit ordinal   amounts to a computable sequence of notations for smaller ordinals limiting to  . Any limit notation   is of the form   where the  '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  .

Although   implies  ,   does not imply  .

In order for  ,   must be reachable from   by repeatedly applying the operations   and   for  . In other words,   when   is eventually referenced in the definition of   given by  .

A Computably Enumerable Order Extending the Kleene Order

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For arbitrary  , say that   when   is reachable from   by repeatedly applying the operations   and   for  . The relation   agrees with   on  , but   is computably enumerable: if  , then a computer program will eventually find a proof of this fact.

For any notation  , all   are themselves notations in  .

For  ,   is a notation of   only when all of the criteria below are met:

  • For all  ,   is either  , a power of  , or   for some  .
  • For any  , if   then   is total and strictly increasing under  ; i.e.   for any  .
  • The set   is well-founded, so that there are no infinite descending sequences  .

Basic properties of <O

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  • 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 computable 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 computable functions, the ordinal   is evidently countable.
  • The ordinals with a notation in Kleene's   are exactly the computable ordinals. (The fact that every computable ordinal has a notation follows from the closure of this system of ordinal notations under successor and effective limits.)
  •   is not computably enumerable, but there is a computably enumerable relation which agrees with   precisely on members of  .
  • For any notation  , the set   of notations below   is computably enumerable. However, Kleene's  , when taken as a whole, is   (see analytical hierarchy) and not arithmetical because of the following:
  •   is  -complete (i.e.   is   and every   set is Turing reducible to it)[2] and every   subset of   is effectively bounded in   (a result of Spector).
  • In fact, any   set is many-one reducible to  .[2]
  •   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 computable function   such that if   is an index for a computable well-ordering, then   is a member of   and   is order-isomorphic to an initial segment of the set  .
  • There is a computable function  , which, for members of  , mimics ordinal addition and has the property that  . (Jockusch)

Properties of paths in <O

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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 computably enumerable (c.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-c.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 c.e. degree  , there is a member   of   such that the path   has many-one degree  . In fact, for each computable ordinal  , a notation   exists with  . (Jockusch)
  • There exist   paths through   which are  . Given a progression of computably 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 c.e., but computable. (Jockusch)
  • Various other paths in   have been shown to exist, each with specific kinds of reducibility properties. (See references below)

See also

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References

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  1. ^ S. G. Simpson, The Hierarchy Based on the Jump Operator, p.269. The Kleene Symposium (North-Holland, 1980)
  2. ^ a b Ash, Knight, *Computable Structures and the Hyperarithmetical Hierarchy* p.83. Studies in Logic and the Foundations of Mathematics vol. 144 (2000), ISBN 0-444-50072-3.
  • Church, Alonzo (1938), "The constructive second number class", Bull. Amer. Math. Soc., 44 (4): 224–232, doi:10.1090/S0002-9904-1938-06720-1
  • Kleene, S. C. (1938), "On Notation for Ordinal Numbers", The Journal of Symbolic Logic, 3 (4), Association for Symbolic Logic: 150–155, doi:10.2307/2267778, JSTOR 2267778, S2CID 34314018
  • Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Effective Computability, First MIT press paperback edition, ISBN 978-0-262-68052-3
  • Feferman, Solomon; Spector, Clifford (1962), "Incompleteness along paths in progressions of theories", Journal of Symbolic Logic, 27 (4): 383–390, doi:10.2307/2964544, JSTOR 2964544, S2CID 33892829