Extending the system T0 of explicit mathematics: the limit and Mahlo axioms

Jäger, Gerhard; Studer, Thomas

doi:10.1016/s0168-0072(01)00076-8

Cited by 23 publications

(34 citation statements)

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“…EETJ adds types on the second order level, and axiomatize elementary comprehension and join as type construction operations. We dispense here with a detailed description of EETJ which can be found in many papers on Explicit Mathematics (e.g., [JKS01], [JS02] or [Kah07]). Let us just briefly address the finite axiomatization of elementary comprehension and join.…”

Section: Explicit Mathematicsmentioning

confidence: 99%

An Extended Predicative Definition of the Mahlo Universe

Setzer

2010

Ways of Proof Theory

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In this article we develop a Mahlo universe in Explicit Mathematics using extended predicative methods. Our approach differs from the usual construction in type theory, where the Mahlo universe has a constructor that refers to all total functions from families of sets in the Mahlo universe into itself; such a construction is, in the absence of a further analysis, impredicative. By extended predicative methods we mean that universes are constructed from below, even if they have impredicative characteristics.

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Section: Explicit Mathematicsmentioning

confidence: 99%

An Extended Predicative Definition of the Mahlo Universe

Setzer

2010

Ways of Proof Theory

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“…Impredicative Mahlo in explicit mathematics is obtained by adding the principle of inductive generation to EMA, cf. Jager and Studer [12]. §4.…”

Section: W"))mentioning

confidence: 99%

“…An extension of Feferman's theory T 0 by Mahlo axioms is studied in Jager and Studer [12]. Setzer [18] presents a related formulation in the framework of Martin-L6f type theory.…”

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confidence: 99%

Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory

Jäger

Strahm

2001

J. symb. log.

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Abstract. In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established. §1. Introduction. In classical set theory an ordinal K is called a Mahlo ordinal if it is a regular cardinal and if, for every normal function / from K to K, there exists a regular cardinal fi less than K so that {/(£) : £ < ju} C [i. The statement that there exists a Mahlo ordinal is a powerful set existence axiom going beyond theories like ZFC. It also outgrows the existence of inaccessible cardinals, hyper inaccessibles, hyperhyperinaccessible and the like.There is also an obvious recursive analogue of Mahlo ordinal. Typically, an ordinal a is baptized recursively Mahlo, if it is admissible and reflects every TI2 sentence on a smaller admissible ordinal. The corresponding formal theory KPM has been proof-theoretically analyzed by Rathjen [14, 15]. KPM is a highly impredicative theory, and its proof-theoretic strength is significantly beyond that of KPi, the second order theory (A 2 -CA) + (Bl) and Feferman's theory To, which are all proof-theoretically equivalent (cf. [3,6,10]).This article can be seen as a further contribution to the general program of metapredicativity. We have studied other metapredicative theories in Jager, Kahle, Setzer and Strahm [8], Jager and Strahm [11], and Strahm [21, 20]; there also some further background material can be found.One aim here is to look at metapredicative Mahlo in admissible set theory. The corresponding theory, named KPm°, is admissible set theory above the natural numbers as urelements plus n2 reflection on the admissibles. As induction principles we have complete induction on the natural numbers for sets, but do not include e induction.A further aim of this paper is to introduce the concept of Mahloness into explicit mathematics and to analyze the proof-theoretic strength of its metapredicative version. An extension of Feferman's theory T 0 by Mahlo axioms is studied in Jager and Studer [12]. Setzer [18] presents a related formulation in the framework of Martin-L6f type theory.For the formalization of Mahlo in explicit mathematics we work over the basic theory EETJ which comprises the axioms of applicative theories and has type

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“…On this line, some extensions by small large cardinal notions have been introduced and investigated in Marzetta [28], Strahm [39,40], Jäger, Kahle and Studer [19], Jäger and Studer [24], Jäger and Strahm [21,23] and Jäger [18], as similar extensions of Kripke-Platek set theory, of constructive Zermelo-Fraenkel set theory and of Martin-Löf type theory have been considered in the right direction to strengthen these mathematical frameworks and also to develop further proof-theoretic investigations. Also for the extensions by monotone inductive definition, another direction of development, Takahashi [41], Rathjen [30][31][32]35] and Glaß, Rathjen and Schlüter [14] all formulated the systems on classical logic, though Rathjen [32] mentioned extending the result for the intuitionistic version as a further problem which was treated by Tupailo [46].…”

Section: Introductionmentioning

confidence: 99%

A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0

Sato

2015

Annals of Pure and Applied Logic

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We partially solve a long-standing problem in the proof theory of explicit mathematics or the proof theory in general. Namely, we give a lower bound of Feferman's system T 0 of explicit mathematics (but only when formulated on classical logic) with a concrete interpretation of the subsystem Σ 1 2 -AC + (BI) of second order arithmetic inside T 0 . Whereas a lower bound proof in the sense of proof-theoretic reducibility or of ordinal analysis was already given in 80s, the lower bound in the sense of interpretability we give here is new. We apply the new interpretation method developed by the author and Zumbrunnen [37], which can be seen as the third kind of model construction method for classical theories, after Cohen's forcing and Krivine's classical realizability. It gives us an interpretation between classical theories, by composing interpretations between intuitionistic theories.

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Extending the system T0 of explicit mathematics: the limit and Mahlo axioms

Cited by 23 publications

References 20 publications

An Extended Predicative Definition of the Mahlo Universe

An Extended Predicative Definition of the Mahlo Universe

Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory

A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0

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