Universes in Type Theory Part II – Autonomous Mahlo

We introduce the autonomous Mahlo universe which is an extension of Martin-Löf type theory which we consider as predicatively justified and which has a strength which goes substantially beyond that of the Mahlo universe, which is before writing this paper the strongest predicatively justified published extension of Martin-Löf type theory. We conjecture it to have the same proof theoretic strength as Kripke-Platek set theory extended by one recursively autonomous Mahlo ordinal and finitely many admissibles above it. Here a recursively autonomous Mahlo universe ordinal is an ordinal κ which is recursively hyperα-Mahlo for all α < κ. We introduce as well as intermediate steps the hyper-Mahlo and hyperα-Mahlo universes, and give meaning explanations for these theories as well as for the super and the Mahlo universe. We introduce a model for the autonomous Mahlo universe, and determine an upper bound for its proof theoretic strength, therefore establishing one half of the conjecture mentioned before. The autonomous Mahlo universe is the crucial intermediate step for understanding the Π3-reflecting universe, which will be published in a successor of this article and which is even stronger and will slightly exceed the strength of Kripke-Platek set theory plus the principle of Π3-reflection.