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Indexed InductionRecursion
, 2001
"... We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in ..."
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Cited by 51 (17 self)
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We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductiverecursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.
Inductionrecursion and initial algebras
 Annals of Pure and Applied Logic
, 2003
"... 1 Introduction Inductionrecursion is a powerful definition method in intuitionistic type theory in the sense of Scott ("Constructive Validity") [31] and MartinL"of [17, 18, 19]. The first occurrence of formal inductionrecursion is MartinL"of's definition ..."
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Cited by 33 (12 self)
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1 Introduction Inductionrecursion is a powerful definition method in intuitionistic type theory in the sense of Scott (&quot;Constructive Validity&quot;) [31] and MartinL&quot;of [17, 18, 19]. The first occurrence of formal inductionrecursion is MartinL&quot;of's definition of a universe `a la Tarski [19], which consists of a set U
Universes in Type Theory Part II – Autonomous Mahlo
, 2009
"... We introduce the autonomous Mahlo universe which is an extension of MartinLöf 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 MartinLöf 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 MartinLöf type theory. We conjecture it to have the same proof theoretic strength as KripkePlatek 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 αMahlo for all α < κ. We introduce as well as intermediate steps the hyperMahlo and hyper αMahlo 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 Π3reflecting universe, which will be published in a successor of this article and which is even stronger and will slightly exceed the strength of KripkePlatek set theory plus the principle of Π3reflection. 1