Kleene's O

In set theory and computability theory, Kleene's ${\displaystyle {\mathcal {O}}}$ 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, ${\displaystyle \omega _{1}^{CK}}$. Since ${\displaystyle \omega _{1}^{CK}}$ is the first ordinal not representable in a computable system of ordinal notations the elements of ${\displaystyle {\mathcal {O}}}$ 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 ${\displaystyle {\mathcal {O}}}$; 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.

Kleene's ${\displaystyle {\mathcal {O}}}$

The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner. For members ${\displaystyle p}$ of ${\displaystyle {\mathcal {O}}}$, the ordinal for which ${\displaystyle p}$ is a notation is ${\displaystyle |p|}$. ${\displaystyle {\mathcal {O}}}$ and ${\displaystyle <_{\mathcal {O}}}$ (a partial ordering of Kleene's ${\displaystyle {\mathcal {O}}}$) are the smallest sets such that the following holds.

• The natural number 0 belongs to Kleene's ${\displaystyle {\mathcal {O}}}$ and ${\displaystyle |0|=0}$.
• If ${\displaystyle i}$ belongs to Kleene's ${\displaystyle {\mathcal {O}}}$ and ${\displaystyle |i|=\alpha }$, then ${\displaystyle 2^{i}}$ belongs to Kleene's ${\displaystyle {\mathcal {O}}}$ and ${\displaystyle |2^{i}|=\alpha +1}$ and ${\displaystyle i<_{\mathcal {O}}2^{i}}$.
• Suppose ${\displaystyle \{e\}}$ is the ${\displaystyle e}$-th partial recursive function. If ${\displaystyle e}$ is total, with range contained in ${\displaystyle {\mathcal {O}}}$, and for every natural number ${\displaystyle n}$, we have ${\displaystyle \{e\}(n)<_{\mathcal {O}}\{e\}(n+1)}$, then ${\displaystyle 3\cdot 5^{e}}$ belongs to Kleene's ${\displaystyle {\mathcal {O}}}$, ${\displaystyle \{e\}(n)<_{\mathcal {O}}3\cdot 5^{e}}$ for each ${\displaystyle n}$ and ${\displaystyle |3\cdot 5^{e}|=\lim _{k}|\{e\}(k)|}$, i.e. ${\displaystyle 3\cdot 5^{e}}$ is a notation for the limit of the ordinals ${\displaystyle \gamma _{k}}$ where ${\displaystyle |\{e\}(k)|=\gamma _{k}}$ for every natural number ${\displaystyle k}$.
• ${\displaystyle p<_{\mathcal {O}}q}$ and ${\displaystyle q<_{\mathcal {O}}r}$ imply ${\displaystyle p<_{\mathcal {O}}r}$ (this guarantees that ${\displaystyle <_{\mathcal {O}}}$ is transitive.)

This definition has the advantages that one can recursively enumerate the predecessors of a given ordinal (though not in the ${\displaystyle <_{\mathcal {O}}}$ ordering) and that the notations are downward closed, i.e., if there is a notation for ${\displaystyle \gamma }$ and ${\displaystyle \alpha <\gamma }$ then there is a notation for ${\displaystyle \alpha }$.

Basic properties of ${\displaystyle <_{\mathcal {O}}}$

• If ${\displaystyle |i|=\alpha }$ and ${\displaystyle |j|=\beta }$ and ${\displaystyle i<_{\mathcal {O}}j\,,}$ then ${\displaystyle \alpha <\beta }$; but the converse may fail to hold.
• ${\displaystyle <_{\mathcal {O}}}$ induces a tree structure on ${\displaystyle {\mathcal {O}}}$, so ${\displaystyle {\mathcal {O}}}$ is well-founded.
• ${\displaystyle {\mathcal {O}}}$ only branches at limit ordinals; and at each notation of a limit ordinal, ${\displaystyle {\mathcal {O}}}$ is infinitely branching.
• Since every recursive function has countably many indices, each infinite ordinal receives countably many notations; the finite ordinals have unique notations, ${\displaystyle n}$ usually denoted ${\displaystyle n_{\mathcal {O}}}$.
• The first ordinal that doesn't receive a notation is called the Church–Kleene ordinal and is denoted by ${\displaystyle \omega _{1}^{CK}}$. Since there are only countably many recursive functions, the ordinal ${\displaystyle \omega _{1}^{CK}}$ is evidently countable.
• The ordinals with a notation in Kleene's ${\displaystyle {\mathcal {O}}}$ 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.)
• ${\displaystyle <_{\mathcal {O}}}$ is not recursively enumerable, but there is a recursively enumerable relation which agrees with ${\displaystyle <_{\mathcal {O}}}$ precisely on members of ${\displaystyle {\mathcal {O}}}$.
• For any notation ${\displaystyle p}$, the set ${\displaystyle \lbrace q\mid q<_{\mathcal {O}}p\rbrace }$ of notations below ${\displaystyle p}$ is recursively enumerable. However, Kleene's ${\displaystyle {\mathcal {O}}}$, when taken as a whole, is ${\displaystyle \Pi _{1}^{1}}$ (see analytical hierarchy).
• In fact, ${\displaystyle {\mathcal {O}}}$ is ${\displaystyle \Pi _{1}^{1}}$-complete and every ${\displaystyle \Sigma _{1}^{1}}$ subset of ${\displaystyle {\mathcal {O}}}$ is effectively bounded in ${\displaystyle {\mathcal {O}}}$ (a result of Spector).
• ${\displaystyle {\mathcal {O}}}$ 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 recursive function ${\displaystyle f}$ such that if ${\displaystyle e}$ is an index for a recursive well-ordering, then ${\displaystyle f(e)}$ is a member of ${\displaystyle {\mathcal {O}}}$ and ${\displaystyle <_{e}}$ is order-isomorphic to an initial segment of the set ${\displaystyle \{p\mid p<_{\mathcal {O}}f(e)\}}$.
• There is a recursive function ${\displaystyle +_{\mathcal {O}}}$, which, for members of ${\displaystyle {\mathcal {O}}}$, mimics ordinal addition and has the property that ${\displaystyle \max\{p,q\}\leq p+_{\mathcal {O}}q}$. (Jockusch)

Properties of paths in ${\displaystyle {\mathcal {O}}}$

A path in ${\displaystyle {\mathcal {O}}}$ is a subset ${\displaystyle {\mathcal {P}}}$ of ${\displaystyle {\mathcal {O}}}$ which is totally ordered by ${\displaystyle <_{\mathcal {O}}}$ and is closed under predecessors, i.e. if ${\displaystyle p}$ is a member of a path ${\displaystyle {\mathcal {P}}}$ and ${\displaystyle q<_{\mathcal {O}}p}$ then ${\displaystyle q}$ is also a member of ${\displaystyle {\mathcal {P}}}$. A path ${\displaystyle {\mathcal {P}}}$ is maximal if there is no element of ${\displaystyle {\mathcal {O}}}$ which is above (in the sense of ${\displaystyle <_{\mathcal {O}}}$) every member of ${\displaystyle {\mathcal {P}}}$, otherwise ${\displaystyle {\mathcal {P}}}$ is non-maximal.

• A path ${\displaystyle {\mathcal {P}}}$ is non-maximal if and only if ${\displaystyle {\mathcal {P}}}$ is recursively enumerable (r.e.). It follows by remarks above that every element ${\displaystyle p}$ of ${\displaystyle {\mathcal {O}}}$ determines a non-maximal path ${\displaystyle {\mathcal {P}}}$; and every non-maximal path is so determined.
• There are ${\displaystyle 2^{\aleph _{0}}}$ maximal paths through ${\displaystyle {\mathcal {O}}}$; since they are maximal, they are non-r.e.
• In fact, there are ${\displaystyle 2^{\aleph _{0}}}$ maximal paths within ${\displaystyle {\mathcal {O}}}$ of length ${\displaystyle \omega ^{2}}$. (Crossley, Schütte)
• For every non-zero ordinal ${\displaystyle \lambda <\omega _{1}^{CK}}$, there are ${\displaystyle 2^{\aleph _{0}}}$ maximal paths within ${\displaystyle {\mathcal {O}}}$ of length ${\displaystyle \omega ^{2}\cdot \lambda }$. (Aczel)
• Further, if ${\displaystyle {\mathcal {P}}}$ is a path whose length is not a multiple of ${\displaystyle \omega ^{2}}$ then ${\displaystyle {\mathcal {P}}}$ is not maximal. (Aczel)
• For each r.e. degree ${\displaystyle d}$, there is a member ${\displaystyle e_{d}}$ of ${\displaystyle {\mathcal {O}}}$ such that the path ${\displaystyle {\mathcal {P}}=\lbrace p\mid p<_{\mathcal {O}}e_{d}\rbrace }$ has many-one degree ${\displaystyle d}$. In fact, for each recursive ordinal ${\displaystyle \alpha \geq \omega ^{2}}$, a notation ${\displaystyle e_{d}}$ exists with ${\displaystyle |e_{d}|=\alpha }$. (Jockusch)
• There exist ${\displaystyle \aleph _{0}}$ paths through ${\displaystyle {\mathcal {O}}}$ which are ${\displaystyle \Pi _{1}^{1}}$. Given a progression of recursively enumerable theories based on iterating Uniform Reflection, each such path is incomplete with respect to the set of true ${\displaystyle \Pi _{1}^{0}}$ sentences. (Feferman & Spector)
• There exist ${\displaystyle \Pi _{1}^{1}}$ paths through ${\displaystyle {\mathcal {O}}}$ each initial segment of which is not merely r.e., but recursive. (Jockusch)
• Various other paths in ${\displaystyle {\mathcal {O}}}$ have been shown to exist, each with specific kinds of reducibility properties. (See references below)

See also

• Recursive ordinal
• Large countable ordinal
• Ordinal notation

References

• 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, Association for Symbolic Logic, 3 (4): 150–155, doi:10.2307/2267778, JSTOR 2267778
• 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