Ordinal notation

One could continue in this way, but it would give us an infinite number of functions. So instead let us merge the unary functions 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.

The function 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=ω^ε).

One has 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 ω^ω. "ξξξ0000" for ${\displaystyle \omega ^{\omega ^{\omega }}.}$

In general, ξ(0,β) = β+1. While ξ(1+α,β) = ω^ωα·(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 ω^ωα+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 that enumerate epsilon numbers, then they will be able to name any ordinal less than the first epsilon number that 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).

"Exponential polynomials" in 0 and ω gives a system of ordinal notation for ordinals less than ε₀. 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.

Ackermann (1951) described a system of ordinal notation rather weaker than the system described earlier by Veblen. The limit of his system is sometimes called the Ackermann ordinal.

Bachmann (1950) harvtxt error: no target: CITEREFBachmann1950 (help) 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 (1987) described a very powerful system of ordinal notation called "ordinal diagrams", which is hard to understand but was later simplified by Feferman.

Feferman introduced theta functions, described in Buchholz (1986) as follows. For an ordinal α, θ_α is a function mapping ordinals to ordinals. Often θ_α(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, ω₁, ω₂, ..., ω_ω, together with the ordinals less than β by the operations of ordinal addition and the functions θ_ξ for ξ<α. And the function θ_γ is defined to be the function enumerating the ordinals δ with δ∉C(γ,δ). The problem with this system is that ordinal notations and collapsing functions are not identical, and therefore this function does not qualify as an ordinal notation. An associated ordinal notation is not known.

Buchholz (1986) described the following system of ordinal notation as a simplification of Feferman's theta functions. Define:

• Ω_ξ = ω_ξ if ξ > 0, Ω₀ = 1

The functions ψ_v(α) for α an ordinal, v an ordinal at most ω, are defined by induction on α as follows:

• ψ_v(α) is the smallest ordinal not in C_v(α)

where C_v(α) is the smallest set such that

• C_v(α) contains all ordinals less than Ω_v
• C_v(α) is closed under ordinal addition
• C_v(α) is closed under the functions ψ_u (for u≤ω) applied to arguments less than α.

This system has about the same strength as Fefermans system, as ${\displaystyle \theta \varepsilon _{\Omega _{v}+1}0=\psi _{0}(\varepsilon _{\Omega _{v}+1})}$ for v ≤ ω. Yet, while this system is powerful, it does not qualify as an ordinal notation. Buchholz did create an associated ordinal notation, yet it is complicated: the definition is in the main article.

Kleene (1938) described a system of notation for all recursive ordinals (those less than the Church–Kleene ordinal). 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 that 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 that contains one element for each smaller ordinal and is effectively ordered. Kleene's ${\displaystyle {\mathcal {O}}}$ denotes a canonical (and very non-computable) set of notations. It uses a subset of the natural numbers instead of finite strings of symbols, and is not recursive, therefore, once again, not qualifying as an ordinal notation.