Wellordering proofs for metapredicative Mahlo

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 KPm0 of admissible set theory, transfinite induction along initial segments of the ordinal φω00, for φ being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in the paper Jäger and Strahm [11] are indeed shar

Transfinite dependent choice and ω-model reflection

In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, ω-model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences among the introduced subsystems. In particular we prove the equivalence of Σ1 1 transfinite dependent choice and Π2 1 reflection on ω-models of Σ1 1-D

On the relationship between fixed points and iteration in admissible set theory without foundation

Abstract.: In this article we show how to use the result in Jäger and Probst [7] to adapt the technique of pseudo-hierarchies and its use in Avigad [1] to subsystems of set theory without foundation. We prove that the theory KPi0 of admissible sets without foundation, extended by the principle (Σ-FP), asserting the existence of fixed points of monotone Σ operators, has the same proof-theoretic ordinal as KPi0 extended by the principle (Σ-TR), that allows to iterate Σ operations along ordinals. By Jäger and Probst [6] we conclude that the metapredicative Mahlo ordinal φω00 is also the ordinal of KPi0+(Σ-FP). Hence the relationship between fixed points and iteration persists in the framework of set theory without foundatio

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

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 establishe

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

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

The extended predicative Mahlo universe in Martin-Lof type theory

This paper addresses the long-standing question of the predicativity of the Mahlo universe. A solution, called the extended predicative Mahlo universe, has been proposed by Kahle and Setzer in the context of explicit mathematics. It makes use of the collection of untyped terms (denoting partial functions) which are directly available in explicit mathematics but not in Martin-Lof type theory. In this paper, we overcome the obstacle of not having direct access to untyped terms in Martin-Lof type theory by formalizing explicit mathematics with an extended predicative Mahlo universe in Martin-Lof type theory with certain indexed inductive-recursive definitions. In this way, we can relate the predicativity question to the fundamental semantics of Martin-Lof type theory in terms of computation to canonical form. As a result, we get the first extended predicative definition of a Mahlo universe in Martin-Lof type theory. To this end, we first define an external variant of Kahle and Setzer\u27s internal extended predicative universe in explicit mathematics. This is then formalized in Martin-Lof type theory, where it becomes an internal extended predicative Mahlo universe. Although we make use of indexed inductive-recursive definitions that go beyond the type theory $\mathbf {IIRD}$ of indexed inductive-recursive definitions defined in previous work by the authors, we argue that they are constructive and predicative in Martin-Lof\u27s sense. The model construction has been type-checked in the proof assistant Agda

Wellordering proofs for metapredicative Mahlo

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 KPm0 of admissible set theory, transfinite induction along initial segments of the ordinal φω00, for φ being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in the paper Jäger and Strahm [11] are indeed sharp

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

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 T0 of explicit mathematics (but only when formulated on classical logic) with a concrete interpretat ion of the subsystem Σ12-AC+ (BI) of second order arithmetic inside T0. Whereas a lower bound proof in the sense of proof-theoretic reducibility or of ordinalanalysis 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 (2015), 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