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 (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available.
Contents
- Ordinal notations
- Relationship to systems of arithmetic
- Predicative definitions and the Veblen hierarchy
- The FefermanSchtte ordinal and beyond
- Impredicative ordinals
- Unrecursable recursive ordinals
- The ChurchKleene ordinal
- Admissible ordinals
- Beyond admissible ordinals
- Unprovable ordinals
- A pseudo well ordering
- References
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 (not to be confused with the first uncountable ordinal, ω1), described below. Ordinal numbers below ω1CK are the recursive ordinals (see below). 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 (descriptions). Larger and larger ordinals can be defined, but they become more and more difficult to describe.
Ordinal notations
Recursive ordinals (or computable ordinals) 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 (i.e., computable) 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 (Turing machine, say) can manipulate them (and, essentially, compare them).
A different definition uses Kleene's system of ordinal notations. Briefly, an ordinal notation is either the name zero (describing the ordinal 0), or the successor of an ordinal notation (describing the successor of the ordinal described by that notation), or a Turing machine (computable function) that produces an increasing sequence of ordinal notations (that describe the ordinal that is the limit of the sequence), and ordinal notations are (partially) ordered so as to make the successor of o greater than o and to make the limit greater than any term of the sequence (this order is computable; however, the set O of ordinal notations itself is highly non-recursive, owing to the impossibility of deciding whether a given Turing machine does indeed produce a sequence of notations); 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 (countable) ordinal, the Church-Kleene ordinal (see below).
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 (the limit of the simplest program that enumerates all natural numbers).
Relationship to systems of arithmetic
There is a relation between computable ordinals and certain formal systems (containing arithmetic, that is, at least a reasonable fragment of Peano arithmetic).
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 (or beyond) ε0: while the ordinal ε0 can easily be arithmetically described (it is countable), 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 (a theorem by Gentzen), so by Gödel's second incompleteness theorem, Peano's axioms cannot formalize that reasoning. (This is at the basis of the Kirby–Paris theorem on Goodstein sequences.) 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 (see below), 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.
Predicative definitions and the Veblen hierarchy
We have already mentioned (see Cantor normal form) the ordinal ε0, which is the smallest satisfying the equation
More generally, the
Ordering:
The Feferman–Schütte ordinal and beyond
The smallest ordinal such that
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
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 (in an ad hoc manner), and different extensions and variations of it were described by Buchholz, Takeuti (ordinal diagrams), Feferman (θ systems), 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(α)=σ for any ordinal α satisfying σ≤α≤Ω. However the fact that we included Ω allows us to get past this point: ψ(Ω+1) 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 (sometimes just called the Howard ordinal, ψ(εΩ+1) with the notation above) 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. (Strictly speaking it is not known that all of these really are ordinals: by construction, the ordinal strength of a theory can only be proved to be an ordinal from an even stronger theory. So for the large cardinal axioms this becomes quite unclear.)
The Church–Kleene ordinal
The set of recursive ordinals is an ordinal that is the smallest ordinal that cannot be described in a recursive way. (It is not the order type of any recursive well-ordering of the integers.) That ordinal is a countable ordinal called the Church–Kleene ordinal,
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 (described above) 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
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
Beyond admissible ordinals
An ordinal that is both admissible and a limit of admissibles, or equivalently such that
An admissible ordinal
“Unprovable” ordinals
We can imagine even larger ordinals that are still countable. For example, if ZFC has a transitive model (a hypothesis stronger than the mere hypothesis of consistency, and implied by the existence of an inaccessible cardinal), then there exists a countable
Even larger countable ordinals, called the stable ordinals, can be defined by indescribability conditions or as those
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