Ordinal Collapsing Function

In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then “collapse” them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals.

The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system “runs out of fuel” and cannot name a certain ordinal, a much larger ordinal is brought “from above” to give a name to that critical point. An example of how this works will be detailed below, for an ordinal collapsing function defining the Bachmann-Howard ordinal (i.e., defining a system of notations up to the Bachmann-Howard ordinal).

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 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 the Greek letter (psi).