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First order theories for nonmonotone inductive definitions: recursively inaccessible and Mahlo
, 1998
"... In this paper first order theories for nonmonotone inductive definitions are introduced, and a prooftheoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given. 1 Introduction Let # be an operator on the power ..."
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In this paper first order theories for nonmonotone inductive definitions are introduced, and a prooftheoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given. 1 Introduction Let # be an operator on the power set P (N) of the natural numbers, i.e. a mapping from P (N) to P (N). Then # can be used to generate subsets I # # of the natural numbers if we define I # # := I <# # # #(I <# # ) and I <# # := # {I # # : # < #} by transfinite recursion on the ordinals. Furthermore we let I # # := # {I # # : # an ordinal } be the set of natural numbers inductively defined by #. Obviously there exists a least ordinal # so that I # # = I <# # . We call this ordinal the closure ordinal of the inductive definition generated by # and know that I # # is identical to I # # . The sets I # # are the stages of the inductive definition generated by #. If K is a class of operators, th...
Proof theory of MartinLöf type theory. An overview
 MATHEMATIQUES ET SCIENCES HUMAINES, 42 ANNÉE, N O 165:59 – 99
, 2004
"... We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert’s programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent ..."
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Cited by 4 (2 self)
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We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert’s programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of MartinLöf type theory with Wtype and one microscopic universe containing only two finite sets in carried out. Then we look at the analysis MartinLöf theory with Wtype and a universe closed under the Wtype, and consider the extension of type theory by one Mahlo universe and its prooftheoretic analysis. Finally we repeat the concept of inductiverecursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.
Universes in Type Theory Part I
, 2005
"... Abstract We give an overview over universes in MartinL"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 KripkePlatek set theory having ..."
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Abstract We give an overview over universes in MartinL&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 KripkePlatek set theory having the same proof theoretic strength. The extensions of KripkePlatek set theory used formalise the existence of a recursively inaccessible ordinal, a recursively hyperinaccessible 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 determining as strong as possible predicatively justified extensions of MartinL&quot;of type theory (MLTT) and to determine their precise proof theoretic strength.We see three main reasons for following this research program:
Universes in Type Theory Part II Autonomous Mahlo and \Pi 3Reflection
, 2005
"... Abstract We introduce an extension of MartinL"of type theory, which we conjecture to have the same proof theoretic strength as KripkePlatek set theory (KP) extended by one \Pi 3reflecting ordinal and finitely many admissibles above it. That would mean that the proof theoretic strength of ..."
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Abstract We introduce an extension of MartinL&quot;of type theory, which we conjecture to have the same proof theoretic strength as KripkePlatek set theory (KP) extended by one \Pi 3reflecting ordinal and finitely many admissibles above it. That would mean that the proof theoretic strength of this type theory is substantially bigger than that of any previous predicatively justified extensions of MartinL&quot;of type theory, including the Mahlo universe. The universe is constructed following the principles of ordinal notation systems of strength KP plus one \Pi 3reflecting ordinal, therefore extracting key ideas of these notation systems. We introduce a model for this type theory, and determine an upper bound for its proof theoretic strength. This article only presents the main ideas of this model construction, full details will be given in a future article. 1 Introduction This article is a step in a research programme of the author with the goal ofintroducing proof theoretically as strong as possible extensions of MartinL&quot;of type theory, which still can be regarded as predicatively justified. (However,because of our lack of expertise in philosophy, we refrain from giving any meaning explanations.) We have three main reasons for following such a researchprogramme: (1) We hope that this approach gives more insights into the development ofordinal theoretic proof theory. Results in the area of proof theory of
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