In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
and
computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since e ...
,
Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
's
is a canonical subset of the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s when regarded as
ordinal notations. It contains
ordinal notation
In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping t ...
s for every
computable ordinal, that is, ordinals below
Church–Kleene ordinal
In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functions.
The Church–Kleene ordinal and varian ...
,
. 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
In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functions.
The Church–Kleene ordinal and varian ...
). It uses a subset of the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s instead of finite strings of symbols. Unfortunately, there is in general no
effective
Effectiveness is the capability of producing a desired result or the ability to produce desired output. When something is deemed effective, it means it has an intended or expected outcome, or produces a deep, vivid impression.
Etymology
The ori ...
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
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an expl ...
) of any two given notations in Kleene's
; and given any notation for an ordinal, there is a
computably enumerable set
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if:
*There is an algorithm such that the ...
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
of
, the ordinal for which
is a notation is
.
and
(a
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
ing of Kleene's
) are the smallest sets such that the following holds.
*
.
*
*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.
[S. G. Simpson]
The Hierarchy Based on the Jump Operator
p.269. ''The Kleene Symposium'' (North-Holland, 1980)
Basic properties of <O
* If
and
and
then
; but the converse may fail to hold.
*
induces a tree structure on
, so
is
well-founded
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s& ...
.
*
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
In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functions.
The Church–Kleene ordinal and varian ...
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
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if:
*There is an algorithm such that the ...
, 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
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers ...
) and not arithmetical because of the following:
* 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 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
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
*
Computable ordinal
*
Large countable ordinal
In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of releva ...
*
Ordinal notation
In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping t ...
References
*
*
*
*{{citation, last1=Feferman, first1=Solomon, last2=Spector, first2=Clifford, title=Incompleteness along paths in progressions of theories, journal=Journal of Symbolic Logic, volume=27, issue=4, year=1962, pages=383–390, doi=10.2307/2964544, jstor=2964544, s2cid=33892829
Ordinal numbers