The use and definition of ordinal collapsing functions is inextricably intertwined with the theory of ordinal analysis, since the large countable ordinals defined and denoted by a given collapse are used to describe the ordinal-theoretic strength of certain formal systems, typically[1][2] subsystems of analysis (such as those seen in the light of reverse mathematics), extensions of Kripke–Platek set theory, Bishop-style systems of constructive mathematics or Martin-Löf-style systems of intuitionistic type theory.
Ordinal collapsing functions are typically denoted using some variation of either the Greek letter
The choice of the ordinal collapsing function given as example below imitates greatly the system introduced by Buchholz[3] but is limited to collapsing one cardinal for clarity of exposition.
in intuitive terms: since the usual operations of addition, multiplication and exponentiation are not sufficient to designate ordinals very far, we attempt to systematically create new names for ordinals by taking the first one which does not have a name yet, and whenever we run out of names, rather than invent them in an ad hoc fashion or using diagonal schemes, we seek them in the ordinals far beyond the ones we are constructing (beyond
representation" is an obvious generalization of the Cantor normal form (which is the case
and repeat the operation until the process terminates (any decreasing sequence of ordinals is finite).
and the various coefficients involved (including as exponents) the pieces of the representation (they are all
, the canonical ordinal notation defined coincides with the iterated Cantor normal form (by definition).
notation (the pieces being themselves written in iterated Cantor normal form): e.g.,
(and write the pieces of that in iterated Cantor normal form): so
Canonicalness can be checked recursively: an expression is canonical if and only if it is either the iterated Cantor normal form of an ordinal less than
The order is checked by lexicographic verification at all levels (keeping in mind that
Actually we will define canonical sequences for certain uncountable ordinals, too, namely the uncountable ordinals of countable cofinality (if we are to hope to define a sequence converging to them...) which are representable (that is, all of whose
It appears as though the expressions are getting more and more complicated whereas, in fact, the ordinals always decrease.
which counts the number of steps of the process before termination if one always selects the
Concerning the second statement, a precise version is given by ordinal analysis: for example, Kripke–Platek set theory can prove[4] that the process terminates for any given
function gives us a system of notations (assuming we can somehow write down all countable ordinals!)
A variation on this scheme, which makes little difference when using just two (or finitely many) collapsing functions, but becomes important for infinitely many of them, is to define i.e., allow the use of
by simultaneous (rather than "downward") induction, and this is important if we are to use infinitely many collapsing functions.
ordinals from the start, he does not need to allow multiplication or exponentiation; also, Buchholz does not introduce the numbers
functions: this makes the entire scheme much more elegant and more concise to define, albeit more difficult to understand.
This system is also sensibly equivalent to the earlier (and much more difficult to grasp) "ordinal diagrams" of Takeuti[5] and
Most definitions of ordinal collapsing functions found in the recent literature differ from the ones we have given in one technical but important way which makes them technically more convenient although intuitively less transparent.
was invented by Heinz Bachmann, somewhat cumbersome as it depends on fundamental sequences for all limit ordinals; and the original definition is complicated.
The limit of this system, sometimes called the Extended Buchholz Ordinal, is much greater, equal to
Chris Bird devised the following shorthand for the extended Veblen function
Jäger's ψ is a hierarchy of single-argument ordinal functions ψκ indexed by uncountable regular cardinals κ smaller than the least weakly Mahlo cardinal M0 introduced by German mathematician Gerhard Jäger in 1984.
This is a sophisticated simplification of Jäger's ψ created by Denis Maksudov.
Then, Rathjen's Ψ function is based on the least weakly compact cardinal to create large countable ordinals.
are defined in mutual recursion in the following way: As noted in the introduction, the use and definition of ordinal collapsing functions is strongly connected with the theory of ordinal analysis, so the collapse of this or that large cardinal must be mentioned simultaneously with the theory for which it provides a proof-theoretic analysis.