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We present well-ordering proofs for Martin-Lof's type theory with W-type 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...

this paper we use methods of proof theory to assign ordinals to classes of (terminating) programs, the idea being that the ordinal assignment provides a uniform way of measuring computational complexity "in the large". We are not concerned with placing prior (e.g. polynomial) bounds on computation--length, but rather with general methods of assessing the complexity of natural classes of programs according to the ways in which they are constructed. We begin with an overview of the method in section 2, the crucial idea being supplied by Buchholz's ! +

Ordinal systems are structures for describing ordinal notation systems, which extend the more predicative approaches to ordinal notation systems, like the Cantor normal form, the Veblen function and the Schutte Klammer symbols, up to the Bachmann-Howard ordinal. oe-ordinal systems, which are natural extensions of this approach, reach without the use of cardinals the strength of the theories for transfinitely iterated inductive definitions ID oe in an essentially predicative way. We explore the relationship with the traditional approach to ordinal notation systems via cardinals and determine, using "extended Schutte Klammer symbols", the exact strength of oe-ordinal systems. 1 Introduction 1.1 Motivation The original problem, which motivated the research in this article, seemed to be a pedagogical one. We have been trying to teach ordinal notation systems above the BachmannHoward ordinal several times. The impression we had was that we were able to teach the technical develo...

A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie " - the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency - technical results in pro...

. We develop an alternative approach to well-ordering proofs beyond the Bachmann-Howard ordinal using transfinite sequences of ordinal notations and use it in order to carry out well-ordering proofs for oe-ordinal 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 Kripke-Platek set theory with one recursivly inaccessible. The upper bound is determined by giving well-ordering proofs, which use the technique of transfinite sequences. We derive from the new approach the traditional approach to well-ordering 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 ...

We introduce a type theory FA # , which has at least the strength of finitely iterated inductive definitions ID<# . This type theory has as ground types trees with finitely many branching degrees (so called free algebras). We introduce an equality in this theory, without the need for undecidable prime formulas. Then we give a direct well-ordering proof for this theory by representing a ordinal denotation system in the iteration of Kleene's O. This can be easily done, by introducing functions on the trees, which correspond to the functions in the ordinal denotation system. The proof shows, that FA # proofs transfinite induction up to D 0 D n 0, which shows, that the strength of FA # is at least ID<# . It seems to be obvious, that this bound is sharp. 1 Definition of the type theory FA # Definition 1.1 The type theory FA # is defined as follows: (a) The ground types are defined inductively by: If n # 0 and # 1 , . . . , # n are ground types, then (# 1 , . . . , # n ) is a ground type. The type (# 1 , . . . , # n ) should be the type of well-founded trees with branching degrees # 1 , . . . , # n . (b) Ground types are types, and if #, # are types then (# # #) is a type. We will omit brackets, using the usual conventions. (c) If # = (# 1 , . . . , # n ) is a ground type, then for i = 1, . . . , n C # i is a constant of type (# i # #) # # (C # i are the constructors for this type) and if # is as before and # a type, then we have the recursion constant R #,# of type ((# 1 # #) # (# 1 # #) # #) # # ((# n # #) # (# n # #) # #) # # # # We will write (C 1 : # 1 , . . . , C n : # n ) for (# 1 , . . . , # n ) to indicate, that C i are names for C # i . 1 (d) The terms are the typed lambda terms, built using the constants of (c). We write t ...

this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinal-theoretic proof theory, which take the place of the original Hilbert Program. Since this part of the talk is now incorporated in the first two sections of the BSL-paper [48] there is no point in reproducing it here. Secondly, we shall omit those parts of the talk concerned with infinitary proof systems of ramified set theory as they can also be found in [48] and even more detailed in [45]. Thirdly, thanks to the aforementioned omissions, the advantage of present paper over the talk is to allow for a much more detailed account of the actual information furnished by ordinal analyses and the role of large cardinal hypotheses in devising ordinal representation systems. 2 Observations on ordinal analyses

We present some new decision and comparison problems of unusually high computational complexity. Most of the problems are strictly combinatorial in nature; others involve basic logical notions. Their complexities range from iterated exponential time completeness to # 0 time completeness to #(# # ,0) time completeness to #(# # ,,0) time completeness. These three ordinals are well known ordinals from proof theory, and their associated complexity classes represent new levels of computational complexity for natural decision problems. Proofs will appear in an extended version of this manuscript to be published elsewhere. 1. Iterated exponential time - universal relational sentences Let F be a function from A* into B*, where A,B are finite alphabets. We say that F is iterated exponential time computable if and only if there is a multitape Turing machine TM (which processes inputs from A* and outputs from B*) and an integer constant c > 0 such that TM computes F(x) with run time at most...

this paper, we omit it here. (a) Let c = D c 0 c 1 ::: cm , a = D a 0 a 1 ::: an with principal terms c 1 ; :::; c m ; a 1 ; :::; an . 1. < : From c m : : : c 1 D c 0 we get by IH o(c m ) : : : o(c 1 ) o(D c 0 ) = o(c 0 ) < +1 and thus o(c) < +1 o(D a 0 ) o(a). 2. = and c 0 a 0 : By IH o(c 0 ) < o(a 0 ). Since D c 0 2 OT, we have G c 0 c 0 and thus by IH o(c 0 ) 2 C(o(c 0 ); o(c 0 )). Hence o(c 0 ) < o(a 0 ) by Theorem 1.2(c). Now o(c) o(a) follows as in 1. (using that o(a) is additively closed). 3. = & c 0 = a 0 & c 1 ::: cm a 1 ::: an : Immediate by IH. (b) 1. c = c 0 ::: c k 1 with k 6= 1: Then G c i a and thus (by IH) o(c i ) 2 C := C(o(a); o(a)) for i < k