Ordinal collapsing function: Difference between revisions

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* Gerhard Jäger and Wolfram Pohlers<ref>Jäger & Pohlers, 1983 (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber.)</ref> 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<ref name="Rathjen-survey"/> to <math>\Delta^1_2</math>-comprehension plus [[bar induction]]. Roughly speaking, this collapse can be obtained by adding the <math>\alpha \mapsto \Omega_\alpha</math> function itself to the list of constructions to which the <math>C(\cdot)</math> collapsing system applies.

* Michael Rathjen<ref>Rathjen, 1991 (Arch. Math. Logic)</ref> 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 authorRathjen<ref>Rathjen, 1994 (Ann. Pure Appl. Logic)</ref> later described the collapse of a [[weakly compact cardinal]] to describe the ordinal-theoretic strength of Kripke-Platek set theory augmented by certain [[reflection principle]]s (concentrating on the case of <math>\Pi_3</math>-reflection). Very roughly speaking, this proceeds by introducing the first cardinal <math>\Xi(\alpha)</math> which is <math>\alpha</math>-hyper-Mahlo and adding the <math>\alpha \mapsto \Xi(\alpha)</math> function itself to the collapsing system.

* Even more recently, the same authorRathjen has begun<ref>Rathjen, 2005 (Arch. Math. Logic)</ref> the investigation of the collapse of yet larger cardinals, with the ultimate goal of achieving an ordinal analysis of <math>\Pi^1_2</math>-comprehension (which is proof-theoretically equivalent to the augmentation of Kripke-Platek by <math>\Sigma_1</math>-separation).

== Notes ==