1 Introduction Induction-recursion is a powerful definition method in intuitionistic type theory in the sense of Scott ("Constructive Validity") [31] and Martin-L"of [17, 18, 19]. The first occurrence of formal induction-recursion is Martin-L"of's definition of a universe `a la Tarski [19], which consists of a set U

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 Martin-Lof'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 Mahlo-universe. Together with [Se96a] it follows jMLMj = /\Omega 1(\Omega M+! ). 1 Introduction An ordinal M is recursively Mahlo iff M is admissible and every M-recursive 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. ...

This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and Martin-Lof's intuitionistic theory of types. This paper treats Mahlo's -numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders. Finally the theorems of that extension of CZF are interpreted in an extension of Martin-Lof's intuitionistic theory of types by a universe. 1 Prefatory and historical remarks The paper is organized as follows: After recalling Mahlo's -numbers and relating the history of universes in Martin-Lof type theory in section 1, we study notions of inaccessibility in the context of Aczel's constructive set theo...

The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. This leads to consistency proofs for the theories KP + Πn–reflection using a small amount of arithmetic (PRA) and the well–foundedness of a certain ordinal notation system with respect to primitive recursive descending sequences. Regarding future work, we intend to avail ourselves of these new cut elimination techniques to attain an ordinal analysis of Π 1 2 comprehension by approaching Π1 2 comprehension through transfinite levels of reflection. 1

This article will survey the state of the art nowadays, in particular recent advance in proof theory beyond admissible proof theory, giving some prospects of success of obtaining an ordinal analysis of \Pi

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 define a realizability interpretation of Aczel's Constructive Set Theory CZF into Explicit Mathematics. The final results are that CZF extended by Mahlo principles is realizable in corresponding extensions of T0 , thus providing relative lower bounds for the proof-theoretic strength of the latter. Introduction Several di#erent frameworks have been founded in the 70-es aiming to give a foundation for constructive mathematics. The most well-developed of them nowadays are Martin-Lof type theory, Aczel's constructive set theory, and Feferman's explicit mathematics. While constructive set theory was built to have an immediate type interpretation, no theory stronger than # 1 2 -CA, which proof-theoretically is still far below the basic system T 0 of Explicit Mathematics, have been shown up to now to be directly embeddable into explicit systems. It also yielded that the only method for establishing lower bounds for T 0 and its extensions remained to be well-ordering proofs. This omissi...

In this paper it is shown that constructive set theory with an axiom asserting the existence of a Mahlo set can be embedded into Setzer's Mahlo type theory. 1 Mahloness in constructive set theory Definition 1.1 A set M is said to be Mahlo if M is set-inaccessible and for every R 2 mv( M M) there exists a set-inaccessible I 2 M such that 8x 2 I 9y 2 I hx; yi 2 R: Lemma 1.2 If M is Mahlo and R 2 mv( M M), then for every a 2 M there exists a set-inaccessible I 2 M such that a 2 I and 8x 2 I 9y 2 I hx; yi 2 R: Proof : Set S := fhx; ha; yii : hx; yi 2 Rg. Then S 2 mv( M M) too. Hence there exists I 2 M such that 8x 2 I 9y 2 I hx; yi 2 S. Now pick c 2 I. Then hc; di 2 S for some d 2 I. Moreover, d = ha; yi for some y. In particular, a 2 I. Further, for each x 2 I there exists u 2 I such that hx; ui 2 S. As a result, u = ha; yi and hx; yi 2 R for some y. Since u 2 I implies y 2 I, the latter shows that 8x 2 I 9y 2 I hx; yi 2 R. ut Definition 1.3 Let Reg s (A) be the stateme...

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

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