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.

"Unrecursable" recursive ordinals

By dropping the requirement of having a concrete description, even larger recursive countable ordinals can be obtained as the ordinals measuring the strengths of various strong theories; roughly speaking, these ordinals are the smallest order types of "natural" ordinal notations that the theories cannot prove are well ordered. By taking stronger and stronger theories such as second-order arithmetic, Zermelo set theory, Zermelo–Fraenkel set theory, or Zermelo–Fraenkel set theory with various large cardinal axioms, one gets some extremely large recursive ordinals.

Beyond recursive ordinals

The Church–Kleene ordinal

The supremum of the set of recursive ordinals is the smallest ordinal that cannot be described in a recursive way. That ordinal is a countable ordinal called the Church–Kleene ordinal,. Thus, is the smallest non-recursive ordinal, and there is no hope of precisely "describing" any ordinals from this point on—we can only define them. But it is still far less than the first uncountable ordinal,. However, as its symbol suggests, it behaves in many ways rather like. For instance, one can define ordinal collapsing functions using instead of.

Admissible ordinals

The Church–Kleene ordinal is again related to Kripke–Platek set theory, but now in a different way: whereas the Bachmann–Howard ordinal was the smallest ordinal for which KP does not prove transfinite induction, the Church–Kleene ordinal is the smallest α such that the construction of the Gödel universe, L, up to stage α, yields a model of KP. Such ordinals are called admissible, thus is the smallest admissible ordinal.
By a theorem of Friedman (mathematician)|Friedman], Jensen, and Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal but for Turing machines with oracles. One sometimes writes for the -th ordinal that is either admissible or a limit of smaller admissibles.

Beyond admissible ordinals

is the smallest limit of admissible ordinals, yet the ordinal itself is not admissible. It is also the smallest such that is a model of -comprehension.
An ordinal that is both admissible and a limit of admissibles, or equivalently such that is the -th admissible ordinal, is called recursively inaccessible, and the least recursively inaccessible may be denoted. An ordinal that is both recursively inaccessible and a limit of recursively inaccessibles is called recursively hyperinaccessible. There exists a theory of large ordinals in this manner that is highly parallel to that of large cardinals. For example, we can define recursively Mahlo ordinals: these are the such that every -recursive closed unbounded subset of contains an admissible ordinal. The 1-section of Harrington's functional is equal to, where is the least recursively Mahlo ordinal.^p.171
But note that we are still talking about possibly countable ordinals here.

Reflection

For a set of formulae, a limit ordinal is called -reflecting if the rank satisfies a certain reflection property for each -formula. These ordinals appear in ordinal analysis of theories such as KP+Π₃-ref, a theory augmenting Kripke-Platek set theory by a -reflection schema. They can also be considered "recursive analogues" of some uncountable cardinals such as weakly compact cardinals and indescribable cardinals. For example, an ordinal which -reflecting is called recursively weakly compact. For finite, the least -reflecting ordinal is also the supremum of the closure ordinals of monotonic inductive definitions whose graphs are Π_m+1⁰.
In particular, -reflecting ordinals also have a characterization using higher-type functionals on ordinal functions, lending them the name 2-admissible ordinals. An unpublished paper by Solomon Feferman supplies, for each finite, a similar property corresponding to -reflection.

Nonprojectibility

An admissible ordinal is called nonprojectible if there is no total -recursive injective function mapping into a smaller ordinal. Being nonprojectible is a much stronger condition than being admissible, recursively inaccessible, or even recursively Mahlo. By Jensen's method of projecta, this statement is equivalent to the statement that the Gödel universe, L, up to stage α, yields a model of KP + -separation. However, -separation on its own is not a strong enough axiom schema to imply nonprojectibility, in fact there are transitive models of +-separation of any countable admissible height.
Nonprojectible ordinals are tied to Jensen's work on projecta. The least ordinals that are nonprojectible relative to a given set are tied to Harrington's construction of the smallest reflecting Spector 2-class.^p.174

"Unprovable" ordinals

We can imagine even larger ordinals that are still countable. For example, if ZFC has a transitive model, then there exists a countable such that is a model of ZFC. Such ordinals are beyond the strength of ZFC in the sense that it cannot prove their existence.
If is a recursively enumerable set theory consistent with V=''L'', then the least such that is less than the least stable ordinal, which follows.

Stable ordinals

Even larger countable ordinals, called the stable ordinals, can be defined by indescribability conditions or as those such that is a Σ₁-elementary submodel of L; the existence of these ordinals can be proved in ZFC, and they are closely related to the nonprojectible ordinals from a model-theoretic perspective. For countable, stability of is equivalent to.
The least stable level of has some definability-related properties. Letting be least such that :

A set has a definition in iff it is a member of.^p.6
A set is iff it is a member of.^p.6
A set is iff it is -recursively enumerable, in the terminology of alpha recursion theory.^p.6

Variants of stable ordinals

These are weakened variants of stable ordinals. There are ordinals with these properties smaller than the aforementioned least [|nonprojectible ordinal], for example an ordinal is -stable iff it is -reflecting for all natural.

A countable ordinal is called -stable iff
A countable ordinal is called -stable iff, where is the least admissible ordinal larger than.
A countable ordinal is called -stable iff, where is the least admissible ordinal larger than an admissible ordinal larger than.
A countable ordinal is called inaccessibly-stable iff, where is the least recursively inaccessible ordinal larger than.
A countable ordinal is called Mahlo-stable iff, where is the least recursively Mahlo ordinal larger than.
A countable ordinal is called doubly -stable iff there is a -stable ordinal such that.

Stronger weakenings of stability have appeared in proof-theoretic publications, including analysis of subsystems of second-order arithmetic.

A pseudo-well-ordering

Within the scheme of notations of Kleene some represent ordinals and some do not. One can define a recursive total ordering that is a subset of the Kleene notations and has an initial segment which is well-ordered with order-type. Every recursively enumerable nonempty subset of this total ordering has a least element. So it resembles a well-ordering in some respects. For example, one can define the arithmetic operations on it. Yet it is not possible to effectively determine exactly where the initial well-ordered part ends and the part lacking a least element begins.
For an example of a recursive pseudo-well-ordering, let S be ATR₀ or another recursively axiomatizable theory that has an ω-model but no hyperarithmetical ω-models, and conservatively extend S with Skolem functions. Let T be the tree of finite partial ω-models of S: A sequence of natural numbers is in T iff S plus ∃m φ ⇒ φ has no inconsistency proof shorter than n. Then the Kleene–Brouwer order of T is a recursive pseudowellordering.
Any such construction must have order type, where is the order type of, and is a recursive ordinal.

On recursive ordinals

Wolfram Pohlers, Proof theory, Springer 1989 . This is probably the most readable book on large countable ordinals.
Gaisi Takeuti, Proof theory, 2nd edition 1987
Kurt Schütte, Proof theory, Springer 1977
Craig Smorynski, The varieties of arboreal experience Math. Intelligencer 4, no. 4, 182–189; contains an informal description of the Veblen hierarchy.
Hartley Rogers Jr., Theory of Recursive Functions and Effective Computability McGraw-Hill
Larry W. Miller, Normal Functions and Constructive Ordinal Notations, The Journal of Symbolic Logic, volume 41, number 2, June 1976, pages 439 to 459,,
Hilbert Levitz, , expository article
Herman Ruge Jervell, , manuscript in progress.

Beyond recursive ordinals

Both recursive and nonrecursive ordinals

Michael Rathjen, "The realm of ordinal analysis." in S. B. Cooper and J. Truss : Sets and Proofs. 219–279. At .