Large countable ordinal

In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations. However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not ; various more-concrete ways of defining ordinals that definitely have notations are available.
Since there are only countably many notations, all ordinals with notations are exhausted well [|below] the first uncountable ordinal ω₁; their supremum is called Church–Kleene ω₁ or ω, described below. Ordinal numbers below ω are the recursive ordinals. Countable ordinals larger than this may still be defined, but do not have notations.
Due to the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted. The ordinals described here are not as large as the ones described in large cardinals, but they are large among those that have constructive notations. Larger and larger ordinals can be defined, but they become more and more difficult to describe.

Generalities on recursive ordinals

Ordinal notations

s are certain countable ordinals: loosely speaking those represented by a computable function. There are several equivalent definitions of this: the simplest is to say that a computable ordinal is the order-type of some recursive well-ordering of the natural numbers; so, essentially, an ordinal is recursive when we can present the set of smaller ordinals in such a way that a computer can manipulate them.
A different definition uses Kleene's system of ordinal notations. Briefly, an ordinal notation is either the name zero, or the successor of an ordinal notation, or a Turing machine that produces an increasing sequence of ordinal notations, and ordinal notations are ordered so as to make the successor of o greater than o and to make the limit greater than any term of the sequence ; a recursive ordinal is then an ordinal described by some ordinal notation.
Any ordinal smaller than a recursive ordinal is itself recursive, so the set of all recursive ordinals forms a certain ordinal, the Church–Kleene ordinal.
It is tempting to forget about ordinal notations, and only speak of the recursive ordinals themselves: and some statements are made about recursive ordinals which, in fact, concern the notations for these ordinals. This leads to difficulties, however, as even the smallest infinite ordinal, ω, has many notations, some of which cannot be proved to be equivalent to the obvious notation.

Relationship to systems of arithmetic

There is a relation between computable ordinals and certain formal systems.
Certain computable ordinals are so large that while they can be given by a certain ordinal notation o, a given formal system might not be sufficiently powerful to show that o is, indeed, an ordinal notation: the system does not show transfinite induction for such large ordinals.
For example, the usual first-order Peano axioms do not prove transfinite induction for ε₀: while the ordinal ε₀ can easily be arithmetically described, the Peano axioms are not strong enough to show that it is indeed an ordinal; in fact, transfinite induction on ε₀ proves the consistency of Peano's axioms, so by Gödel's second incompleteness theorem, Peano's axioms cannot formalize that reasoning. Since Peano arithmetic can prove that any ordinal less than ε₀ is well ordered, we say that ε₀ measures the proof-theoretic strength of Peano's axioms.
But we can do this for systems far beyond Peano's axioms. For example, the proof-theoretic strength of Kripke–Platek set theory is the Bachmann–Howard ordinal, and, in fact, merely adding to Peano's axioms the axioms that state the well-ordering of all ordinals below the Bachmann–Howard ordinal is sufficient to obtain all arithmetical consequences of Kripke–Platek set theory.

Specific recursive ordinals

Predicative definitions and the Veblen hierarchy

We have already mentioned the ordinal ε₀, which is the smallest satisfying the equation, so it is the limit of the sequence 0, 1,,,,... The next ordinal satisfying this equation is called ε₁: it is the limit of the sequence
More generally, the -th ordinal such that is called. We could define as the smallest ordinal such that, but since the Greek alphabet does not have transfinitely many letters it is better to use a more robust notation: define ordinals by transfinite induction as follows: let and let be the -th fixed point of , and when is a limit ordinal, define as the -th common fixed point of the for all. This family of functions is known as the Veblen hierarchy . is called the Veblen function.
Ordering: if and only if either or or.

The Feferman–Schütte ordinal and beyond

The smallest ordinal such that is known as the Feferman–Schütte ordinal and generally written. It can be described as the set of all ordinals that can be written as finite expressions, starting from zero, using only the Veblen hierarchy and addition. The Feferman–Schütte ordinal is important because, in a sense that is complicated to make precise, it is the smallest ordinal that cannot be described using smaller ordinals. It measures the strength of such systems as "arithmetical transfinite recursion".
More generally, Γ_α enumerates the ordinals that cannot be obtained from smaller ordinals using addition and the Veblen functions.
It is, of course, possible to describe ordinals beyond the Feferman–Schütte ordinal. One could continue to seek fixed points in a more and more complicated manner: enumerate the fixed points of, then enumerate the fixed points of that, and so on, and then look for the first ordinal α such that α is obtained in α steps of this process, and continue diagonalizing in this ad hoc manner. This leads to the definition of the "small" and "large" Veblen ordinals.

Impredicative ordinals

To go far beyond the Feferman–Schütte ordinal, one needs to introduce new methods. Unfortunately there is not yet any standard way to do this: every author in the subject seems to have invented their own system of notation, and it is quite hard to translate between the different systems. The first such system was introduced by Bachmann in 1950, and different extensions and variations of it were described by Buchholz, Takeuti, Feferman, Aczel, Bridge, Schütte, and Pohlers. However most systems use the same basic idea, of constructing new countable ordinals by using the existence of certain uncountable ordinals. Here is an example of such a definition, described in much greater detail in the article on ordinal collapsing function:

ψ is defined to be the smallest ordinal that cannot be constructed by starting with 0, 1, ω and Ω, and repeatedly applying addition, multiplication and exponentiation, and ψ to previously constructed ordinals.

Here Ω = ω₁ is the first uncountable ordinal. It is put in because otherwise the function ψ gets "stuck" at the smallest ordinal σ such that ε_σ=σ: in particular ψ=σ for any ordinal α satisfying σ≤''α≤Ω. However the fact that we included Ω allows us to get past this point: ψ is greater than σ''. The key property of Ω that we used is that it is greater than any ordinal produced by ψ.
To construct still larger ordinals, we can extend the definition of ψ by throwing in more ways of constructing uncountable ordinals. There are several ways to do this, described to some extent in the article on ordinal collapsing function.
The Bachmann–Howard ordinal is an important one, because it describes the proof-theoretic strength of Kripke–Platek set theory. Indeed, the main importance of these large ordinals, and the reason to describe them, is their relation to certain formal systems as explained above. However, such powerful formal systems as full second-order arithmetic, let alone Zermelo–Fraenkel set theory, seem beyond reach for the moment.

Beyond even the Bachmann-Howard ordinal

Beyond this, there are multiple recursive ordinals which aren't as well known as the previous ones. The first of these is Buchholz's ordinal, defined as, abbreviated as just, using the previous notation. It is the proof-theoretic ordinal of, a first-order theory of arithmetic allowing quantification over the natural numbers as well as sets of natural numbers, and, the "formal theory of finitely iterated inductive definitions".
Since the hydras from Buchholz's hydra game are isomorphic to Buchholz's ordinal notation, the ordinals up to this point can be expressed using hydras from the game.^p.136 For example corresponds to.
Next is the Takeuti-Feferman-Buchholz ordinal, the proof-theoretic ordinal of ; and another subsystem of second-order arithmetic: - comprehension + transfinite induction, and, the "formal theory of -times iterated inductive definitions". In this notation, it is defined as. It is the supremum of the range of Buchholz's psi functions. It was first named by David Madore.
The next ordinal is mentioned in a piece of code describing , and defined by "AndrasKovacs" as.
The next ordinal is mentioned in the same piece of code as earlier, and defined as. It is the proof-theoretic ordinal of.
This next ordinal is, once again, mentioned in this same piece of code, defined as, is the proof-theoretic ordinal of. In general, the proof-theoretic ordinal of is equal to — note that in this certain instance, represents, the first nonzero ordinal.
Next is an unnamed ordinal, referred by David Madore as the "countable" collapse of, where is the first inaccessible cardinal. This is the proof-theoretic ordinal of Kripke-Platek set theory augmented by the recursive inaccessibility of the class of ordinals, or, on the arithmetical side, of -comprehension + transfinite induction. Its value is equal to using an unknown function.
Next is another unnamed ordinal, referred by David Madore as the "countable" collapse of, where is the first Mahlo cardinal. This is the proof-theoretic ordinal of KPM, an extension of Kripke-Platek set theory based on a Mahlo cardinal. Its value is equal to using one of Buchholz's various psi functions.
Next is another unnamed ordinal, referred by David Madore as the "countable" collapse of, where is the first weakly compact cardinal. This is the proof-theoretic ordinal of Kripke-Platek set theory + Π3 - Ref. Its value is equal to using Rathjen's Psi function.
Next is another unnamed ordinal, referred by David Madore as the "countable" collapse of, where is the first -indescribable cardinal. This is the proof-theoretic ordinal of Kripke-Platek set theory + Πω-Ref. Its value is equal to using Stegert's Psi function, where =.
Next is the last unnamed ordinal, referred by David Madore as the proof-theoretic ordinal of Stability. This is the proof-theoretic ordinal of Stability, an extension of Kripke-Platek set theory. Its value is equal to using Stegert's Psi function, where =.
Next is a group of ordinals which not that much are known about, but are still fairly significant :

The proof-theoretic ordinal of second-order arithmetic.
A possible limit of Taranovsky's C ordinal notation.
The proof-theoretic ordinal of ZFC.