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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
Extending MartinLof type theory by one Mahlouniverse
 Arch. Math. Logic, 39:155
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A model for a type theory with Mahlo universe
, 1996
"... We present a type theory T T M, extending MartinLöf Type Theory by adding one Mahlo universe V, a universe being the type theoretic analogue of one recursive Mahlo ordinal. A model, formulated in a KripkePlatek style set theory KP M +, is given and we show that the proof theoretical strength of T ..."
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Cited by 8 (6 self)
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We present a type theory T T M, extending MartinLöf Type Theory by adding one Mahlo universe V, a universe being the type theoretic analogue of one recursive Mahlo ordinal. A model, formulated in a KripkePlatek style set theory KP M +, is given and we show that the proof theoretical strength of T T M is ≤ KP M +  = ψΩ1 (ΩM+ω). By [Se96a], this bound is sharp. 1