A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers.
If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers.
A recursive ordinal notation must satisfy the following two additional properties: There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, Wilfried Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jäger, Isles, Pfeiffer, Wolfram Pohlers, Kurt Schütte, Gaisi Takeuti (called ordinal diagrams), Oswald Veblen.
In many systems, such as Veblen's well known system, the functions are normal functions, that is, they are strictly increasing and continuous in at least one of their arguments, and increasing in other arguments.
As usual, we must start off with a constant symbol for zero, "0", which we may consider to be a function of arity zero.
One could continue in this way, but it would give us an infinite number of functions.
It always gives values larger than its arguments and its range is all ordinals other than 0 and the epsilon numbers (ε=ωε).
The ξ-notations can be used to name any ordinal less than ε0 with an alphabet of only two symbols ("0" and "ξ").
If these notations are extended by adding functions that enumerate epsilon numbers, then they will be able to name any ordinal less than the first epsilon number that cannot be named by the added functions.
There are many equivalent ways to write these; instead of exponential polynomials, one can use rooted trees, or nested parentheses, or the system described above.
His original system was rather cumbersome to use as it required choosing a special sequence converging to each ordinal.
Later systems of notation introduced by Feferman and others avoided this complication.
Buchholz (1986) described the following system of ordinal notation as a simplification of Feferman's theta functions.
Buchholz did create an associated ordinal notation, yet it is complicated: the definition is in the main article.
It uses a subset of the natural numbers instead of finite strings of symbols, and is not recursive, therefore, once again, not qualifying as an ordinal notation.