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**1 - 4**of**4**### Universes in Type Theory Part I-

, 2005

"... Abstract We give an overview over universes in Martin-L"of type theory and consider the following universe constructions: a simple universe, E. Palmgren's super universe and the Mahlo universe. We then introduce models for these theories in extensions of Kripke-Platek set theory having ..."

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Abstract We give an overview over universes in Martin-L&quot;of type theory and consider the following universe constructions: a simple universe, E. Palmgren's super universe and the Mahlo universe. We then introduce models for these theories in extensions of Kripke-Platek set theory having the same proof theoretic strength. The extensions of Kripke-Platek set theory used formalise the existence of a recursively inaccessible ordinal, a recursively hyper-inaccessible ordinal, and a recursively Mahlo ordinal. Using these models we determine upper bounds for the proof theoretic strength of the theories in questions. In case of simple universes and the Mahlo universe, these bounds have been shown by the author to be sharp. This article is an overview over the main techniques in developing these models, full details will be presented in a series of future articles. 1 Introduction This article presents some results of a research program with the goal of de-termining as strong as possible predicatively justified extensions of Martin-L&quot;of type theory (MLTT) and to determine their precise proof theoretic strength.We see three main reasons for following this research program:

### Indexed Induction-Recursion

"... An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductive-recursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function from that set into another type. An indexed inductive-recursive ..."

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An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductive-recursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function from that set into another type. An indexed inductive-recursive definition (IIRD) is a combination of both. We present a closed theory which allows us to introduce all IIRDs in a natural way without much encoding. By specialising it we also get a closed theory of IID. Our theory of IIRDs includes essentially all definitions of sets which occur in Martin-Löf type theory. We show in particular that Martin-Löf’s computability predicates for dependent types and Palmgren’s higher order universes are special kinds of IIRD and thereby clarify why they are constructively acceptable notions. We give two axiomatisations. The first and more restricted one formalises a principle for introducing meaningful IIRD by using the data-construct in the original version of the proof assistant Agda for Martin-Löf type theory. The second one admits a more general form of introduction rule, including the introduction rule for the intensional identity relation, which is not covered by the restricted one. If we add an extensional identity relation to our logical framework, we show that the theories of restricted and general IIRD are equivalent by interpreting them in each other. Finally, we show the consistency of our theories by constructing a model in classical set theory extended by a Mahlo cardinal.

### SPECIFYING AND REASONING IN THE CALCULUS OF OBJECTS

, 2005

"... Since type theory merges constructive logic with functional programming language it appears a very promising system for formal program construction. The present thesis deals with this idea in the environment of a type theoretic system equipped with the type constructor representing a simple form of ..."

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Since type theory merges constructive logic with functional programming language it appears a very promising system for formal program construction. The present thesis deals with this idea in the environment of a type theoretic system equipped with the type constructor representing a simple form of objects. The presented interpretation of an object type requires rather non-trivial extension of the underlaying calculus of the type theory. The notion of box, which is the content containment structure with a marker variable, is added to the syntax of the calculus that allows to delimit internal states of objects in the definitions of methods. The acquired overall simple notation is a justification of the need for this extraordinary extension. The objectives of this thesis are mainly to give the formal representation of simple object model and to show its significant metatheoretical properties. We also show the capabilities of the system for specifying and reasoning about programs by defining the basic concepts of program specifications and several stewise refinement operations and techniques for reasoning about the correctness of programs. To do this the Core Calculus of Objects (CCO) is introduced. CCO is derived from

### Universes in Type Theory Part II – Autonomous Mahlo

, 2009

"... 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 publis ..."

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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 hyper α-Mahlo for all α < κ. We introduce as well as intermediate steps the hyper-Mahlo and hyper α-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. 1