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Wellordering proofs for MartinLöf Type Theory
 Annals of Pure and Applied Logic
, 1998
"... We present wellordering proofs for MartinLof's type theory with Wtype and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is ..."
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Cited by 18 (11 self)
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We present wellordering proofs for MartinLof's type theory with Wtype and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is slightly more than the strength of Feferman's theory T 0 , classical set theory KPI and the subsystem of analysis (# 1 2 CA)+(BI). The strength of intensional and extensional version, of the version a la Tarski and a la Russell are shown to be the same. 0 Introduction 0.1 Proof theory and Type Theory Proof theory and type theory have been two answers of mathematical logic to the crisis of the foundations of mathematics at the beginning of the century. Proof theory was originally established by Hilbert in order to prove the consistency of theories by using finitary methods. When Godel showed that Hilbert's program cannot be carried out as originally intended, the focus of proof theory ch...
Extending MartinLöf Type Theory by One MahloUniverse
 Arch. Math. Log., 39:155
, 1998
"... We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjens theory KPM. This is achieved by replacing the universe in MartinLof's Type Theory by a new universe V, which has the property that for every function f , mapping families of sets in V to families of set ..."
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Cited by 15 (8 self)
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We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjens theory KPM. This is achieved by replacing the universe in MartinLof's Type Theory by a new universe V, which has the property that for every function f , mapping families of sets in V to families of sets in V, there exists a universe closed under f . We show that the proof theoretical strength of MLM is /\Omega 1\Omega M+! . Therefore we reach a strength slightly greater than jKPMj and V can be considered as a Mahlouniverse. Together with [Se96a] it follows jMLMj = /\Omega 1(\Omega M+! ). 1 Introduction An ordinal M is recursively Mahlo iff M is admissible and every Mrecursive closed unbounded subset of M contains an admissible ordinal. Equivalently, this is the case iff M is admissible and for all \Delta 0 formulas OE(x; y; ~z), and all ~z 2 LM such that 8x 2 LM :9y 2 LM :OE(x; y; ~z) there exists an admissible ordinal fi ! M such that 8x 2 L fi 9y 2 L fi :OE(x; y; ~z) holds. ...
Ordinal Systems, Part 2: One Inaccessible
 In Logic Colloquium ’98
, 2000
"... . We develop an alternative approach to wellordering proofs beyond the BachmannHoward ordinal using transfinite sequences of ordinal notations and use it in order to carry out wellordering proofs for oeordinal systems. We extend the approach of ordinal systems as an alternative way of presentin ..."
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Cited by 4 (1 self)
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. We develop an alternative approach to wellordering proofs beyond the BachmannHoward ordinal using transfinite sequences of ordinal notations and use it in order to carry out wellordering proofs for oeordinal systems. We extend the approach of ordinal systems as an alternative way of presenting ordinal notation systems started in [Set98b] and develop ordinal systems, which have in the limit exactly the strength of KripkePlatek set theory with one recursivly inaccessible. The upper bound is determined by giving wellordering proofs, which use the technique of transfinite sequences. We derive from the new approach the traditional approach to wellordering proofs using distinguished sets. The lower bound is determined by extending the concept of ordinal function generators in [Set98b] to inaccessibles. 1 Introduction This article is a followup of [Set98b]. In that article we introduced ordinal systems as an alternative way of describing ordinal notation systems which usually make ...