Results 1  10
of
22
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 of a universe `a la T ..."
Abstract

Cited by 28 (11 self)
 Add to MetaCart
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 of a universe `a la Tarski [19], which consists of a set U
The Strength of Some MartinLöf Type Theories
 Arch. Math. Logic
, 1994
"... One objective of this paper is the determination of the prooftheoretic strength of Martin Lof's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely ..."
Abstract

Cited by 25 (5 self)
 Add to MetaCart
One objective of this paper is the determination of the prooftheoretic strength of Martin Lof's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with \Delta 1 2 comprehension and bar induction. As MartinLof intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the prooftheoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman. 0 Introduction MartinLof's intuitionistic theory of types was originally introduce...
Inaccessibility in Constructive Set Theory and Type Theory
, 1998
"... 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 MartinLof's intuitionistic theory of types. This paper treats Mahlo's numbers whi ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
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 MartinLof'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 MartinLof'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 MartinLof type theory in section 1, we study notions of inaccessibility in the context of Aczel's constructive set theo...
The Strength of Some MartinLöf Type Theories
 ARCHIVE FOR MATHEMATICAL LOGIC
, 1994
"... One objective of this paper is the determination of the prooftheoretic strength of MartinLöf's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely th ..."
Abstract

Cited by 14 (10 self)
 Add to MetaCart
One objective of this paper is the determination of the prooftheoretic strength of MartinLöf's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with \Delta 1 2 comprehension and bar induction. As MartinLöf intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the prooftheoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman.
Extending the System T_0 of explicit mathematics: the limit and Mahlo axioms
"... In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the socalled limit and Mahlo axioms and present a novel approach to constructing natural recusiontheoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive def ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the socalled limit and Mahlo axioms and present a novel approach to constructing natural recusiontheoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions.
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 1997
"... 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 ma ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
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'. Ordinaltheoretic 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...
An upper bound for the proof theoretical strength of MartinLöf Type Theory with Wtype and one universe
, 1996
"... (2), W (2) and I (3). To make it easier to remember the meaning of the symbols, we give the following hints: r is the (unique) element of an identity type I; n k is the nth element of the finite type N k with k elements, C the Casedistinction for this type; O is the zero, S the Successor, P Primit ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
(2), W (2) and I (3). To make it easier to remember the meaning of the symbols, we give the following hints: r is the (unique) element of an identity type I; n k is the nth element of the finite type N k with k elements, C the Casedistinction for this type; O is the zero, S the Successor, P Primitive recursion or induction over the natural numbers N ; i stands for left inclusion, j for right inclusion, D is the choice in the type A +B of disjoint union of A and B; p 0 and p 1 are the projections, p the pairing for the #typ
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 ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
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
Realization of Constructive Set Theory into Explicit Mathematics: a lower bound for impredicative Mahlo universe
 Transactions American Math. Soc
, 2000
"... 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 prooftheoretic strength of the latter ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
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 prooftheoretic strength of the latter. Introduction Several di#erent frameworks have been founded in the 70es aiming to give a foundation for constructive mathematics. The most welldeveloped of them nowadays are MartinLof 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 prooftheoretically 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 wellordering proofs. This omissi...
Monotone Inductive Definitions in Explicit Mathematics
 Journal of Symbolic Logic
, 1996
"... The context for this paper is Feferman's theory of explicit mathematics, T 0 . We address a problem that was posed in [F 82]. Let MID be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that T 0 + MID, when base ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
The context for this paper is Feferman's theory of explicit mathematics, T 0 . We address a problem that was posed in [F 82]. Let MID be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that T 0 + MID, when based on classical logic, also proves the existence of nonmonotone inductive definitions that arise from arbitrary extensional operations on classifications. From the latter we deduce that MID, when adjoined to classical T 0 , leads to a much stronger theory than T 0 . 1 Introduction Prompted by the question of constructive justification of Spector's consistency proof for analysis, Kreisel initiated in 1963 the study of formal theories featuring inductive definitions (cf. [K 63]). Prooftheoretic investigations (cf. [BFPS 81], [F 82], [Ra 89]) of such theories have shown that the strength of monotone inductive definitions is not greater than that of positive or even accessibility inductive de...