Results 1  10
of
20
Universes in Explicit Mathematics
 Annals of Pure and Applied Logic
, 1999
"... This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathemat ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are prooftheoretically equivalent to Feferman's T 0 . 1 Introduction In some form or another, universes play an important role in many systems of set theory and higher order arithmetic, in various formalizations of constructive mathematics and in logics for computation. One aspect of universes is that they expand the set or type formation principles in a natural and perspicuous way and provide greater expressive power and prooftheoretic strength. The general idea behind universes is quite simple: suppose that we are given a formal system Th comprising certain set (or type) existence principles which are justified on specific philosophical grounds. Then it may be a...
Wellordering proofs for metapredicative Mahlo
 Journal of Symbolic Logic
"... In this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathemati ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
In this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathematics and KPm 0 of admissible set theory, transfinite induction along initial segments of the ordinal ##00, for # being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in the paper Jager and Strahm [11] are indeed sharp. 1 Introduction This paper is a companion to the article Jager and Strahm [11], where systems of explicit mathematics and admissible set theory for metapredicative Mahlo are introduced. Whereas the main concern of [11] was to establish prooftheoretic upper bounds for these systems, in this article we provide the corresponding wellordering proofs, thus showing that the upper bounds derived in [11] are sharp. The central...
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...
Σ 1 1 choice in a theory of sets and classes
"... Dedicated to Wolfram Pohlers on his retirement Several decades ago Friedman showed that the subsystem Σ1 1AC of second order arithmetic is prooftheoretically equivalent – and thus equiconsistent – to (Π1 0CA)<ε0. In this article we prove the analogous result for Σ1 1 choice in the context of the ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Dedicated to Wolfram Pohlers on his retirement Several decades ago Friedman showed that the subsystem Σ1 1AC of second order arithmetic is prooftheoretically equivalent – and thus equiconsistent – to (Π1 0CA)<ε0. In this article we prove the analogous result for Σ1 1 choice in the context of the von NeumannBernaysGödel theory NBG of sets and classes.
The provably terminating operations of the subsystem PETJ of explicit mathematics
, 2010
"... In Spescha and Strahm [15], a system PET of explicit mathematics in the style of Feferman [7, 8] is introduced, and in Spescha and Strahm [16] the addition of the join principle to PET is studied. Changing to intuitionistic logic, it could be shown that the provably terminating operations of PETJ i ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
In Spescha and Strahm [15], a system PET of explicit mathematics in the style of Feferman [7, 8] is introduced, and in Spescha and Strahm [16] the addition of the join principle to PET is studied. Changing to intuitionistic logic, it could be shown that the provably terminating operations of PETJ i are the polytime functions on binary words. However, although strongly conjectured, it remained open whether the same holds true for the corresponding theory PETJ with classical logic. This note supplements a proof of this conjecture. Keywords: Explicit mathematics, polytime functions, nonstandard models
The prooftheoretic analysis of Σ 1 1 transfinite dependent choice
 Annals of Pure and Applied Logic 121 (2003
"... choice. This article provides an ordinal analysis of Σ 1 1 transfinite dependent 1 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
choice. This article provides an ordinal analysis of Σ 1 1 transfinite dependent 1
Autonomous Fixed Point Progressions and Fixed Point Transfinite Recursion
 In Logic Colloquium ’98
"... . This paper is a contribution to the area of metapredicative proof theory. It continues recent investigations on the transfinitely iterated fixed point theories # ID# (cf. [10]) and addresses the question of autonomity in iterated fixed point theories. An external and an internal form of autonom ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
. This paper is a contribution to the area of metapredicative proof theory. It continues recent investigations on the transfinitely iterated fixed point theories # ID# (cf. [10]) and addresses the question of autonomity in iterated fixed point theories. An external and an internal form of autonomous generation of transfinite hierarchies of fixed points of positive arithmetic operators are introduced and prooftheoretically analyzed. This includes the discussion of the principle of socalled fixed point transfinite recursion. Connections to theories for iterated inaccessibility in the context of Kripke Platek set theory without foundation are revealed. 1 Introduction The foundational program to study the principles and ordinals which are implicit in a predicative conception of the universe of sets of natural numbers led to the progression of systems of ramified analysis up to the famous FefermanSchutte ordinal # 0 in the early sixties. Since then numerous theories have been found w...
Metapredicative And Explicit Mahlo: A ProofTheoretic Perspective
"... After briefly discussing the concepts of predicativity, metapredicativity and impredicativity, we turn to the notion of Mahloness as it is treated in various contexts. Afterwards the ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
After briefly discussing the concepts of predicativity, metapredicativity and impredicativity, we turn to the notion of Mahloness as it is treated in various contexts. Afterwards the
Metapredicative Subsystems of Analysis
 Ph.D. thesis, Institut für Informatik und angewandte Mathematik, Univeristät Bern, 2000. & EXPLICIT MAHLO 21
, 2001
"... In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, #model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences of the introduced subsystems. In particular we prove the equivalence of # ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, #model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences of the introduced subsystems. In particular we prove the equivalence of # 1 1 transfinite dependent choice and # 1 2 reflection on #models of # 1 1 DC. 1 Introduction The formal system of classical analysis is second order arithmetic with the full comprehension principle. It was baptized classical analysis, since classical mathematical analysis can be formalized in it. Often, subsystems of classical analysis su#ce as formal framework for particular parts of mathematical analysis. During the last decades a lot of such subsystems have been isolated and prooftheoretically investigated. The subsystems of analysis introduced in this paper belong to metapredicative prooftheory. Metapredicative systems have prooftheoretic ordinals beyond # 0 but can still be tr...