Kleene's O Explained

In set theory and computability theory, Kleene's

l{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,
CK
\omega
1

. Since
CK
\omega
1

is the first ordinal not representable in a computable system of ordinal notations the elements of
l{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

l{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

l{O}

, the ordinal for which

is a notation is

|p|

l{O}

and

<_l{O}

(a partial ordering of Kleene's

l{O}

) are the smallest sets such that the following holds.

0\inl{O}\land|0|=0

i\in{l{O}}\land|i|=\alpha → 2ⁱ\in{l{O}}\land|2^i|=\alpha+1\landi<_l{O

}2^

Suppose

\{e\}

is the

-th partial computable function. If

is total and

rm{range}(\{e\})\subsetl{O}\land\foralln(\{e\}(n)<_l{O

}\(n+1)), then

3 ⋅ 5^e\in{l{O}}\land\foralln,\{e\}(n)<_l{O

} 3 \cdot 5^ \land |3\cdot 5^|=\lim_ | \(k) |

p<_l{O

}q \land q<_r \rightarrow p<_r

This definition has the advantages that one can computably enumerate the predecessors of a given ordinal (though not in the

<_l{O}

ordering) and that the notations are downward closed, i.e., if there is a notation for

\gamma

and

\alpha<\gamma

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.^[1]

Explanation

A member

of Kleene's

l{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

l{O}

, a successor ordinal is defined in terms of a notation for the ordinal immediately preceding it. Specifically, a successor notation

is of the form

2^q

for some other notation

, so that

|p|=|q|+1

The limit notations are those such that

|p|

is a limit ordinal. In Kleene's

l{O}

, a notation for a limit ordinal

\beta

amounts to a computable sequence of notations for smaller ordinals limiting to

\beta

. Any limit notation

is of the form

3 ⋅ 5^e

where the

'th partial computable function

\{e\}:N → N

is a total function listing an infinite sequence

\langleq_0,q_1...\rangle

of notations, which are strictly increasing under the order

<_l{O}

. The limit of the sequence of ordinals

\langle|q_0|,|q_1|...\rangle

|p|

Although

q<_l{O

} p implies

|q|<|p|

does not imply

q<_l{O

} p.

In order for

q<_l{O

} p,

must be reachable from

by repeatedly applying the operations

2^n\mapston

and

3 ⋅ 5^{e\mapsto\{e\}(i)}

for

i\inN

. In other words,

q<_l{O

} p when

is eventually referenced in the definition of

|p|

given by

A Computably Enumerable Order Extending the Kleene Order

For arbitrary

p,q\inN

, say that

q\precp

when

is reachable from

by repeatedly applying the operations

2^n\mapston

and

3 ⋅ 5^{e\mapsto\{e\}(i)}

for

i\indom(\{e\})

. The relation

\prec

agrees with

<_l{O

} on

l{O}

, but

\prec

is computably enumerable: if

q\precp

, then a computer program will eventually find a proof of this fact.

For any notation

p\inl{O}

, all

q\precp

are themselves notations in

l{O}

For

p\inN

is a notation of

l{O}

only when all of the criteria below are met:

For all

q\preceqp

is either

, a power of

, or

3 ⋅ 5^e

for some

e\inN

For any

q\preceqp

, if

q=3 ⋅ 5^e

then

\{e\}

is total and strictly increasing under

\prec

; i.e.

\{e\}(i)\prec\{e\}(i+1)

for any

i\inN

The set

\{q\inN:q\precp\}

is well-founded, so that there are no infinite descending sequences

...\precq_1\precq_0\precp

Basic properties of <_O

|i|=\alpha

and

|j|=\beta

and

i<_l{O

} j \,, then

\alpha<\beta

; but the converse may fail to hold.

<_l{O}

induces a tree structure on

l{O}

, so

l{O}

is well-founded.

l{O}

only branches at limit ordinals; and at each notation of a limit ordinal,

l{O}

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

n_l{O}

The first ordinal that doesn't receive a notation is called the Church–Kleene ordinal and is denoted by

	CK
\omega
	1

. Since there are only countably many computable functions, the ordinal

	CK
\omega
	1

is evidently countable.

The ordinals with a notation in Kleene's

l{O}

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.)

<_l{O}

is not computably enumerable, but there is a computably enumerable relation which agrees with

<_l{O}

precisely on members of

l{O}

For any notation

, the set

\lbraceq\midq<_l{O

} p \rbrace of notations below

is computably enumerable. However, Kleene's

l{O}

, when taken as a whole, is

	1
\Pi
	1

(see analytical hierarchy) and not arithmetical because of the following:

l{O}

	1
\Pi
	1

-complete (i.e.

l{O}

	1
\Pi
	1

and every

	1
\Pi
	1

set is Turing reducible to it)^[2] and every

	1
\Sigma
	1

subset of

l{O}

is effectively bounded in

l{O}

(a result of Spector).

In fact, any

	1
\Pi
	1

set is many-one reducible to

l{O}

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

f(e)

is a member of

l{O}

and

<_e

is order-isomorphic to an initial segment of the set

\{p\midp<_l{O}f(e)\}

There is a computable function

+_l{O}

, which, for members of

l{O}

, mimics ordinal addition and has the property that

max\{p,q\}\leqp+_l{O}q

. (Jockusch)

Properties of paths in <_O

A path in

l{O}

is a subset

l{P}

l{O}

which is totally ordered by

<_l{O}

and is closed under predecessors, i.e. if

is a member of a path

l{P}

and

q<_l{O}p

then

is also a member of

l{P}

. A path

l{P}

is maximal if there is no element of

l{O}

which is above (in the sense of

<_l{O}

) every member of

l{P}

, otherwise

l{P}

is non-maximal.

A path

l{P}

is non-maximal if and only if

l{P}

is computably enumerable (c.e.). It follows by remarks above that every element

l{O}

determines a non-maximal path

l{P}

; and every non-maximal path is so determined.

There are

	\aleph₀
2

maximal paths through

l{O}

; since they are maximal, they are non-c.e.

In fact, there are

	\aleph₀
2

maximal paths within

l{O}

of length

\omega²

. (Crossley, Schütte)

For every non-zero ordinal

λ<

	CK
\omega
	1

, there are

	\aleph₀
2

maximal paths within

l{O}

of length

\omega² ⋅ λ

. (Aczel)

Further, if

l{P}

is a path whose length is not a multiple of

\omega²

then

l{P}

is not maximal. (Aczel)

For each c.e. degree

, there is a member

e_d

l{O}

such that the path

l{P}=\lbracep\midp<_l{O

} e_d \rbrace has many-one degree

. In fact, for each computable ordinal

\alpha\geq\omega²

, a notation

e_d

exists with

|e_d|=\alpha

. (Jockusch)

There exist

\aleph₀

paths through

l{O}

which are

	1
\Pi
	1

. Given a progression of computably enumerable theories based on iterating Uniform Reflection, each such path is incomplete with respect to the set of true

	0
\Pi
	1

sentences. (Feferman & Spector)

There exist

	1
\Pi
	1

paths through

l{O}

each initial segment of which is not merely c.e., but computable. (Jockusch)

Various other paths in

l{O}

have been shown to exist, each with specific kinds of reducibility properties. (See references below)

Notes and References

S. G. Simpson, The Hierarchy Based on the Jump Operator, p.269. The Kleene Symposium (North-Holland, 1980)
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.

Kleene's O Explained

Definition

Explanation

A Computably Enumerable Order Extending the Kleene Order

Basic properties of <O

Properties of paths in <O

See also

Notes and References

Basic properties of <_O

Properties of paths in <_O