Buchholz's psi-functions are a hierarchy of single-argument ordinal functions introduced by German mathematicianWilfried Buchholz in 1986. These functions are a simplified version of the -functions, but nevertheless have the same strength as those. Later on this approach was extended by Jaiger and Schütte.
Definition
Buchholz defined his functions as follows: where and is the set of additive principal numbers in form, the sum of which gives this ordinal : where and Note: Greek letters always denotes ordinals. The limit of this notation is Takeuti–Feferman–Buchholz ordinal.
Properties
Buchholz showed following properties of this functions:
Fundamental sequences and normal form for Buchholz's function
Normal form
The normal form for 0 is 0. If is a nonzero ordinal number then the normal form for is where and and each is also written in normal form.
Fundamental sequences
The fundamental sequence for an ordinal number with cofinality is a strictly increasing sequence with length and with limit, where is the -th element of this sequence. If is a successor ordinal then and the fundamental sequence has only one element. If is a limit ordinal then. For nonzero ordinals, written in normal form, fundamental sequences are defined as follows:
If where then and,
If, then and,
If, then and,
If then and ,
If and then and,
If and then and where.
Explanation
Buchholz is working in Zermelo–Fraenkel set theory, that means every ordinal is equal to set. Then condition means that set includes all ordinals less than in other words. The condition means that set includes:
all ordinals from previous set,
all ordinals that can be obtained by summation the additively principal ordinals from previous set,
all ordinals that can be obtained by applying ordinals less than from the previous set as arguments of functions, where.
That is why we can rewrite this condition as: Thus union of all sets with i.e. denotes the set of all ordinals which can be generated from ordinals by the functions + and, where and. Then is the smallest ordinal that does not belong to this set. Examples Consider the following examples: Then. includes and all possible sums of natural numbers and therefore – first transfinite ordinal, which is greater thanall natural numbers by its definition. includes and all possible sums of them and therefore. If then and. If then and – the smallest epsilon number i.e. first fixed point of. If then and. the second epsilon number, , where denotes the Veblen's function, , where denotes the Feferman's function, Now let's research how works: i.e. includes all countable ordinals. And therefore 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. If then and. For case the set includes functions with all arguments less than i.e. such arguments as and then In the general case: We also can write:
