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.
We have already mentioned (see Cantor normal form) the ordinal ε0, which is the smallest satisfying the equation
The next ordinal satisfying this equation is called ε1: it is the limit of the sequence More generally, the
, but since the Greek alphabet does not have transfinitely many letters it is better to use a more robust notation: define ordinals
This family of functions is known as the Veblen hierarchy (there are inessential variations in the definition, such as letting, for
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 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 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.
There are several ways to do this, described to some extent in the article on ordinal collapsing function.
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.
Next is an unnamed ordinal, referred by David Madore as the "countable" collapse of
Next is another unnamed ordinal, referred by David Madore as the "countable" collapse of
This is the proof-theoretic ordinal of KPM, an extension of Kripke-Platek set theory based on a Mahlo cardinal.
[8] Next is another unnamed ordinal, referred by David Madore as the "countable" collapse of
[9] Next is another unnamed ordinal, referred by David Madore as the "countable" collapse of
[5] This is the proof-theoretic ordinal of Stability, an extension of Kripke-Platek set theory.
is the smallest admissible ordinal (beyond ω in case the axiom of infinity is not included in KP).
contains an admissible ordinal (a recursive analog of the definition of a Mahlo cardinal).
(This is trivially true for regular cardinals; however, we are mainly interested in countable ordinals.)
) is not a strong enough axiom schema to imply nonprojectibility, in fact there are transitive models of
[5][22] The least ordinals that are nonprojectible relative to a given set are tied to Harrington's construction of the smallest reflecting Spector 2-class.
Such ordinals are beyond the strength of ZFC in the sense that it cannot (by construction) prove their existence.
[18] Stronger weakenings of stability have appeared in proof-theoretic publications, including analysis of subsystems of second-order arithmetic.
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 (or even hyperarithmetic) nonempty subset of this total ordering has a least element.
Let T be the tree of (essentially) finite partial ω-models of S: A sequence of natural numbers
[27] Most books describing large countable ordinals are on proof theory, and unfortunately tend to be out of print.