In mathematical logic and set theory, an ordinal notation is a finite sequence of symbols from a finite alphabet which names an ordinal number according to some scheme which gives meaning to the language.
There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, Wilfried Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jäger, Isles, Pfeiffer, Wolfram Pohlers, Kurt Schütte, Gaisi Takeuti (called ordinal diagrams), Oswald Veblen. Given such a scheme, one should be able to define a recursive well-ordering of a subset of the natural numbers by associating a natural number with each finite sequence of symbols via a Gödel numbering. Stephen Cole Kleene has a system of notations, called Kleene's O, which includes ordinal notations but it is not as well behaved as the other systems described here.
Usually one proceeds by defining several functions from ordinals to ordinals and representing each such function by a symbol. In many systems, such as Veblen's well known system, the functions are normal functions, that is, they are strictly increasing and continuous in at least one of their arguments, and increasing in other arguments. Another desirable property for such functions is that the value of the function is greater than each of its arguments, so that an ordinal is always being described in terms of smaller ordinals. There are several such desirable properties. Unfortunately, no one system can have all of them since they contradict each other.
A simplified example using a pairing function 
As usual, we must start off with a constant symbol for zero, "0", which we may consider to be a zero-ary function. This is necessary because there are no smaller ordinals in terms of which zero can be described. The most obvious next step would be to define a unary function, "S", which takes an ordinal to the smallest ordinal greater than it; in other words, S is the successor function. In combination with zero, successor allows one to name any natural number.
The third function might be defined as one which maps each ordinal to the smallest ordinal which cannot yet be described with the above two functions and previous values of this function. This would map β to ω·β except when β is a fixed point of that function plus a finite number in which case one uses ω·(β+1).
The fourth function would map α to ωω·α except when α is a fixed point of that plus a finite number in which case one uses ωω·(α+1).
One could continue in this way, but it would give us an infinite number of functions. So instead let us merge the unary functions together into a binary function. By transfinite recursion on α, we can use transfinite recursion on β to define ξ(α,β) = the smallest ordinal γ such that α < γ and β < γ and γ is not the value of ξ for any smaller α or for the same α with a smaller β.
Thus, define ξ-notations as follows:
- "0" is a ξ-notation for zero.
- If "A" and "B" are replaced by ξ-notations for α and β in "ξAB", then the result is a ξ-notation for ξ(α,β).
- There are no other ξ-notations.
ξ is defined for all pairs of ordinals and is one-to-one. It always gives values larger than its arguments and its range is all ordinals other than 0 and the epsilon numbers (ε=ωε).
ξ(α,β)<ξ(γ,δ) if and only if either (α=γ and β<δ) or (α<γ and β<ξ(γ,δ)) or (α>γ and ξ(α,β)≤δ).
With this definition, the first few ξ-notations are:
- "0" for 0. "ξ00" for 1. "ξ0ξ00" for ξ(0,1)=2. "ξξ000" for ξ(1,0)=ω. "ξ0ξ0ξ00" for 3. "ξ0ξξ000" for ω+1. "ξξ00ξ00" for ω·2. "ξξ0ξ000" for ωω. "ξξξ0000" for
In general, ξ(0,β) = β+1. While ξ(1+α,β) = ωωα·(β+k) for k = 0 or 1 or 2 depending on special situations:
k = 2 if α is an epsilon number and β is finite.
Otherwise, k = 1 if β is a multiple of ωωα+1 plus a finite number.
Otherwise, k = 0.
The ξ-notations can be used to name any ordinal less than ε0 with an alphabet of only two symbols ("0" and "ξ"). If these notations are extended by adding functions which enumerate epsilon numbers, then they will be able to name any ordinal less than the first epsilon number which cannot be named by the added functions. This last property, adding symbols within an initial segment of the ordinals gives names within that segment, is called repleteness (after Solomon Feferman).
Systems of ordinal notation 
There are many different systems for ordinal notation introduced by various authors. It is often quite hard to convert between the different systems.
"Exponential polynomials" in 0 and ω gives a system of ordinal notation for ordinals less than epsilon zero. There are many equivalent ways to write these; instead of exponential polynomials, one can use rooted trees, or nested parentheses, or the system described above.
The 2-variable Veblen functions (Veblen 1908) can be used to give a system of ordinal notation for ordinals less than the Feferman-Schutte ordinal. The Veblen functions in a finite or transfinite number of variables give systems of ordinal notations for ordinals less than the small and large Veblen ordinals.
Bachmann (1950) introduced the key idea of using uncountable ordinals to produce new countable ordinals. His original system was rather cumbersome to use as it required choosing a special sequence converging to each ordinal. Later systems of notation introduced by Feferman and others avoided this complication.
Takeuti (ordinal diagrams) 
Takeuti (1987) described a very powerful system of ordinal notation called "ordinal diagrams", which is hard to understand but was later simplified by Feferman.
Feferman's θ functions 
Feferman introduced theta functions, described in Buchholz (1986) as follows. The function for an ordinal α, θα is a function from ordinals to ordinals. Often θα(β) is written as θαβ. The set C(α,β) is defined by induction on α to be the set of ordinals that can be generated from 0, ℵ1, ℵ2,...,ℵω, together with the ordinals less than β by the operations of ordinal addition and the functions θξ for ξ<α. And the function θγ is defined to be the function enumerating the ordinals δ with δ∉C(γ,δ).
Buchholz (1986) described the following system of ordinal notation as a simplification of Feferman's theta functions. Define:
- Ωξ = ℵξ if ξ > 0, Ω0 = 1
The functions ψv(α) for α an ordinal, v an ordinal at most ω, are defined by induction on α as follows:
- ψv(α) is the smallest ordinal not in Cv(α)
where Cv(α) is the smallest set such that
- Cv(α) contains all ordinals less than Ωv
- Cv(α) is closed under ordinal addition
- Cv(α) is closed under the functions ψu (for u≤ω) applied to arguments less than α.
This system has about the same strength as Fefermans system, as for v ≤ ω.
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, unlike the other systems described above 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. Kleene's denotes a canonical (and very non-computable) set of notations.
See also 
- Ackermann, Wilhelm (1951), "Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse", Math. Z. 53 (5): 403–413, doi:10.1007/BF01175640, MR 0039669
- Buchholz, W. (1986), "A new system of proof-theoretic ordinal functions", Ann. Pure Appl. Logic 32 (3): 195–207, doi:10.1016/0168-0072(86)90052-7, MR 0865989
- "Constructive Ordinal Notation Systems" by Fredrick Gass
- Kleene, S. C. (1938), "On Notation for Ordinal Numbers", The Journal of Symbolic Logic (The Journal of Symbolic Logic, Vol. 3, No. 4) 3 (4): 150–155, doi:10.2307/2267778, JSTOR 2267778
- "Hyperarithmetical Index Sets In Recursion Theory" by Stephen Lempp
- Hilbert Levitz, Transfinite Ordinals and Their Notations: For The Uninitiated, expository article (8 pages, in PostScript)
- Miller, Larry W. (1976), "Normal Functions and Constructive Ordinal Notations", The Journal of Symbolic Logic (The Journal of Symbolic Logic, Vol. 41, No. 2) 41 (2): 439 to 459, doi:10.2307/2272243, JSTOR 2272243
- Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics 1407, Berlin: Springer-Verlag, ISBN 3-540-51842-8, MR 1026933
- Rogers, Hartley (1987) , The Theory of Recursive Functions and Effective Computability, First MIT press paperback edition, ISBN 978-0-262-68052-3
- Schütte, Kurt (1977), Proof theory, Grundlehren der Mathematischen Wissenschaften 225, Berlin-New York: Springer-Verlag, pp. xii+299, ISBN 3-540-07911-4, MR 0505313
- Takeuti, Gaisi (1987), Proof theory, Studies in Logic and the Foundations of Mathematics 81 (Second ed.), Amsterdam: North-Holland Publishing Co., ISBN 0-444-87943-9, MR 0882549
- Veblen, Oswald (1908), "Continuous Increasing Functions of Finite and Transfinite Ordinals", Transactions of the American Mathematical Society (Transactions of the American Mathematical Society, Vol. 9, No. 3) 9 (3): 280–292, doi:10.2307/1988605, JSTOR 1988605