Ordinal Collapsing Function - Collapsing Large Cardinals

Collapsing Large Cardinals

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.

• Gerhard Jäger and Wolfram Pohlers described the collapse of an inaccessible cardinal to describe the ordinal-theoretic strength of Kripke-Platek set theory augmented by the recursive inaccessibility of the class of ordinals (KPi), which is also proof-theoretically equivalent to -comprehension plus bar induction. Roughly speaking, this collapse can be obtained by adding the function itself to the list of constructions to which the collapsing system applies.
• Michael Rathjen then described the collapse of a Mahlo cardinal to describe the ordinal-theoretic strength of Kripke-Platek set theory augmented by the recursive mahloness of the class of ordinals (KPM).
• The same author later described the collapse of a weakly compact cardinal to describe the ordinal-theoretic strength of Kripke-Platek set theory augmented by certain reflection principles (concentrating on the case of -reflection). Very roughly speaking, this proceeds by introducing the first cardinal which is -hyper-Mahlo and adding the function itself to the collapsing system.
• Even more recently, the same author has begun the investigation of the collapse of yet larger cardinals, with the ultimate goal of achieving an ordinal analysis of -comprehension (which is proof-theoretically equivalent to the augmentation of Kripke-Platek by -separation).