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 the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number. If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers. A recursive ordinal notation must satisfy the following two additional properties:

the subset of natural numbers is a recursive set
the induced well-ordering on the subset of natural numbers is a recursive relation

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. 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 function of arity zero. 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 that maps each ordinal to the smallest ordinal that 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 ω^ω·α except when α is a fixed point of that plus a finite number in which case one uses ω^ω·(α+1).

ξ-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

\omega ^{\omega ^{\omega }}.

In general, ξ(0,β) = β+1. While ξ(1+α,β) = ω^{ω^α}·(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^{ω^α+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).

List

There are many different systems for ordinal notation introduced by various authors. It is often quite hard to convert between the different systems.

Cantor

"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.

Veblen

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

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

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 system of ordinal notation known as "ordinal diagrams", whose limit is the Takeuti–Feferman–Buchholz ordinal.^[1] The system was later simplified by Feferman.

Feferman's θ functions

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

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 $\theta \varepsilon _{\Omega _{v}+1}0=\psi _{0}(\varepsilon _{\Omega _{v}+1})$ 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's O

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 ${\mathcal {O}}$ ; 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 ${\mathcal {O}}$ 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.

List of limits of various ordinal notations and collapsing functions

Table of limits of various notations and functions
Notation	Supremum of expressible countable ordinals
Cantor normal form	Epsilon-naught $\varepsilon _{0}$
Binary Veblen function	Feferman-Schütte ordinal $\Gamma _{0}$
Finitary Veblen function	Small Veblen ordinal
Bird's $\theta$ function	Small Veblen ordinal $\theta (\Omega ^{\omega })$
Extended Veblen function	Large Veblen ordinal
Bachmann's $\psi$ function	Bachmann–Howard ordinal $\psi _{\Omega }(\varepsilon _{\Omega +1})$
Madore's $\psi$ function	Bachmann–Howard ordinal $\psi (\varepsilon _{\Omega +1})$
Weiermann's $\vartheta$ function	Bachmann–Howard ordinal $\vartheta (\varepsilon _{\Omega +1})$
Feferman's $\theta$ function	Takeuti–Feferman–Buchholz ordinal $\theta _{\varepsilon _{\Omega _{\omega }+1}}(0)$
Buchholz's $\psi$ function	Takeuti–Feferman–Buchholz ordinal $\psi _{0}(\varepsilon _{\Omega _{\omega }+1})$
Rathjen's $\psi$ function	$\psi _{\Omega }(\chi _{\varepsilon _{M}+1}(0))$
Rathjen's $\Psi$ function	> $\Psi (\varepsilon _{K+1})$
Stegert's First $\Psi$ function	> $\Psi _{(\omega ^{+};{\mathsf {P}}_{0};\epsilon ;\epsilon ;0)}^{\varepsilon _{\Xi }+1}$ ^[2]
Stegert's Second $\Psi$ function	> Large Stegert ordinal $\Psi _{(\omega ^{+};{\mathsf {P}}_{0};\epsilon ;\epsilon ;0)}^{\varepsilon _{Y}+1}$ ^[2]
Taranovsky's $C$ function	Existence unknown, but recursive and very large
Kleene's ${\mathcal {O}}$ function	Church-Kleene ordinal $\omega _{1}^{\mathsf {CK}}$
Klev's ${\mathcal {O}}^{+}$ function	Supremum of writable ordinals $\lambda$
Klev's ${\mathcal {O}}^{++}$ function	Supremum of eventually writable ordinals $\zeta$

References

^ Rathjen, Michael (1 August 2023). "The Art of Measuring the Strength of Theories". Notices of the American Mathematical Society. 70 (7): 1071–1079 – via White Rose.
^ ^a ^b D. Madore, A Zoo of Ordinals (p.2). Accessed 25 October 2021.

Ackermann, Wilhelm (1951), "Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse", Math. Z., 53 (5): 403–413, doi:10.1007/BF01175640, MR 0039669, S2CID 119687180
Bachmann, Heinz (1950), "Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen" (PDF), Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich (in German), 95: 115–147, MR 0036806; English translation by Martin Dowd (2019), arXiv:1903.04609
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, 3 (4): 150–155, doi:10.2307/2267778, JSTOR 2267778, S2CID 34314018
"Hyperarithmetical Index Sets In Recursion Theory" by Steffen Lempp
Hilbert Levitz, Transfinite Ordinals and Their Notations: For The Uninitiated, expository article, 1999 (8 pages, in PostScript)
Miller, Larry W. (1976), "Normal Functions and Constructive Ordinal Notations", The Journal of Symbolic Logic, 41 (2): 439–459, doi:10.2307/2272243, JSTOR 2272243
Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag, doi:10.1007/978-3-540-46825-7, ISBN 978-3-540-51842-6, MR 1026933
Rogers, Hartley (1987) [1967], 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, vol. 225, Berlin-New York: Springer-Verlag, pp. xii+299, ISBN 978-3-540-07911-8, MR 0505313
Takeuti, Gaisi (1987), Proof theory, Studies in Logic and the Foundations of Mathematics, vol. 81 (Second ed.), Amsterdam: North-Holland Publishing Co., ISBN 978-0-444-87943-1, MR 0882549
Veblen, Oswald (1908), "Continuous Increasing Functions of Finite and Transfinite Ordinals", Transactions of the American Mathematical Society, 9 (3): 280–292, doi:10.2307/1988605, JSTOR 1988605

[1] Rathjen, Michael (1 August 2023). "The Art of Measuring the Strength of Theories". Notices of the American Mathematical Society. 70 (7): 1071–1079 – via White Rose.

[stegertzoo-2] D. Madore, A Zoo of Ordinals (p.2). Accessed 25 October 2021.

[1]

[2]

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