Buchholz's psi-functions are a hierarchy of single-argument ordinal functions
introduced by German mathematician Wilfried Buchholz in 1986.
Later on this approach was extended by Jäger[1] and Schütte.
Define: The functions ψv(α) for α an ordinal, v an ordinal at most ω, are defined by induction on α as follows: where Cv(α) is the smallest set such that The limit of this notation is the Takeuti–Feferman–Buchholz ordinal.
be the class of additively principal ordinals.
Buchholz showed following properties of this functions: The normal form for 0 is 0.
The fundamental sequence for an ordinal number
cof ( α ) = β
, written in normal form, fundamental sequences are defined as follows: Buchholz is working in Zermelo–Fraenkel set theory, that means every ordinal
includes: That is why we can rewrite this condition as: Thus union of all sets
is the smallest ordinal that does not belong to this set.
and all possible sums of natural numbers and therefore
– first transfinite ordinal, which is greater than all natural numbers by its definition.
– the smallest epsilon number i.e. first fixed point of
denotes the Veblen function,
includes all possible sums of all countable ordinals and
first uncountable ordinal which is greater than all countable ordinal by its definition i.e. smallest number with cardinality
and then In the general case: We also can write: Buchholz[3] defined an ordinal notation
, braces and commas in the following way: An element of
, a principal term or a braced array of principal terms of length
By convention, every term can be uniquely expressed as either
or a braced, non-empty array of principal terms.
is a recursive strict total ordering on
itself is not a well-ordering, its restriction to a recursive subset
: The normal form for Buchholz's function can be defined by the pull-back of standard form for the ordinal notation associated to it by
is defined in the following way: It is also useful to replace the third case by the following one obtained by combining the second condition: We note that those two formulations are not equivalent.
is a valid formula which is false with respect to the second formulation because of
, while it is a valid formula which is true with respect to the first formulation because of
In this way, the notion of normal form heavily depends on the context.
can be uniquely expressed in normal form for Buchholz's function.