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 ω1CK, described below. Ordinal numbers below ω1CK 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. 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,,,, etc. 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, Γα 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 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:
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 εσ=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.

“Unrecursable” recursive ordinals

By dropping the requirement of having a useful 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 ordinals 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.

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 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 admissibles.

Beyond admissible ordinals

An ordinal that is both admissible and a limit of admissibles, or equivalently such that is the -th admissible ordinal, is called recursively inaccessible. 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. But note that we are still talking about possibly countable ordinals here.
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. It is equivalent to the statement that the Gödel universe, L, up to stage α, yields a model of KP + -separation.

“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.
Even larger countable ordinals, called the stable ordinals, can be defined by indescribability conditions or as those such that is a 1-elementary submodel of L; the existence of these ordinals can be proved in ZFC, and they are closely related to the nonprojectible ordinals.

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.

On recursive ordinals