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

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

An extended predicative definition of the Mahlo universe

This article, which will be reviewed by Zentralblatt Math, contains the first predicative definition of the Mahlo universe, by extending the concept of predicativity. This is a break through result, since it introduces a methodology which allows to justify proof theoretically much stronger theories than were known before predicatively.Before this article predicativity was limited to inductive recursive definition, and it was widely believed that it is impossible to go beyond that notion in a predicative way. With this article for the first time this barrier has been passed using a novel approach

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

The proof-theoretic analysis of transfinitely iterated fixed point theories

This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories and ; the exact proof-theoretic ordinals of these systems are presente

Error and Predicativity

The article surveys ideas emerging within the predicative tradition in the foundations of mathematics, and attempts a reading of predicativity constraints as highlighting different levels of understanding in mathematics. A connection is made with two kinds of error which appear in mathematics: local and foundational errors. The suggestion is that ideas originating in the predicativity debate as a reply to foundational errors are now having profound influence to the way we try to address the issue of local errors. Here fundamental new interactions between computer science and mathematics emerge

The Use of Trustworthy Principles in a Revised Hilbert’s Program

After the failure of Hilbert’s original program due to Gödel’s second incompleteness theorem, relativized Hilbert’s programs have been sug-gested. While most metamathematical investigations are focused on car-rying out mathematical reductions, we claim that in order to give a full substitute for Hilbert’s program, one should not stop with purely mathe-matical investigations, but give an answer to the question why one should believe that all theorems proved in certain mathematical theories are valid. We suggest that, while it is not possible to obtain absolute certainty, it is possible to develop trustworthy core principles using which one can prove the correctness of mathematical theories. Trust can be established by both providing a direct validation of such principles, which is nec-essarily non-mathematical and philosophical in nature, and at the same time testing those principles using metamathematical investigations. We investigate three approaches for trustworthy principles, namely ordinal no-tation systems built from below, Martin-Löf type theory, and Feferman’s system of explicit mathematics. We will review what is known about the strength up to which direct validation can be provided. 1 Reducing Theories to Trustworthy Principles In the early 1920’s Hilbert suggested a program for the foundation of mathemat-ics, which is now called Hilbert’s program. As formulated in [40], “it calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof it-self was to be carried out using only what Hilbert called ’finitary ’ methods. The special epistemological character of finitary reasoning then yields the required justification of classical mathematics. ” Because of Gödel’s second incomplete-ness theorem, Hilbert’s program can be carried out only for very weak theories

Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)

The Workshop "Mathematical Logic: Proof Theory, Constructive Mathematics" focused on proofs both as formal derivations in deductive systems as well as on the extraction of explicit computational content from given proofs in core areas of ordinary mathematics using proof-theoretic methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory

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