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 proof-theoretically 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 proof-theoretic 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...

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 proof-theoretic 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...

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 proof-theoretic strength of the latter. Introduction Several di#erent frameworks have been founded in the 70-es aiming to give a foundation for constructive mathematics. The most well-developed of them nowadays are Martin-Lof 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 proof-theoretically 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 well-ordering proofs. This omissi...

choice. This article provides an ordinal analysis of Σ 1 1 transfinite dependent 1

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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, non-standard models

Dedicated to Wolfram Pohlers on his retirement Several decades ago Friedman showed that the subsystem Σ1 1-AC of second order arithmetic is proof-theoretically equivalent – and thus equiconsistent – to (Π1 0-CA)<ε0. In this article we prove the analogous result for Σ1 1 choice in the context of the von Neumann-Bernays-Gödel theory NBG of sets and classes.

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 proof-theoretically investigated. The subsystems of analysis introduced in this paper belong to metapredicative proof-theory. Metapredicative systems have proof-theoretic ordinals beyond # 0 but can still be tr...

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

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