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Second order theories with ordinals and elementary comprehension

Gerhard Jäger & Thomas Strahm

Archive for Mathematical Logic 34 (6):345-375 (1995)

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(1 other version)Some theories with positive induction of ordinal strength ϕω.Gerhard Jäger & Thomas Strahm - 1996 - Journal of Symbolic Logic 61 (3):818-842.details This paper deals with: (i) the theory ID # 1 which results from $\widehat{\mathrm{ID}}_1$ by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON(μ) plus various forms of positive induction, and (iii) a subtheory of Peano arithmetic with ordinals in which induction on the natural numbers is restricted to formulas which are Σ in the ordinals. We show that these systems have proof-theoretic strength φω 0. Download Export citation Bookmark 8 citations
The [mathematical formula] quantification operator in explicit mathematics with universes and iterated fixed point theories with ordinals.Markus Marzetta & Thomas Strahm - 1997 - Archive for Mathematical Logic 36 (6):391-413.details This paper is about two topics: 1. systems of explicit mathematics with universes and a non-constructive quantification operator $\mu$; 2. iterated fixed point theories with ordinals. We give a proof-theoretic treatment of both families of theories; in particular, ordinal theories are used to get upper bounds for explicit theories with finitely many universes. Download Export citation Bookmark 4 citations
Systems of explicit mathematics with non-constructive μ-operator and join.Thomas Glaß & Thomas Strahm - 1996 - Annals of Pure and Applied Logic 82 (2):193-219.details The aim of this article is to give the proof-theoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the non-constructive μ-operator and join. We make use of standard proof-theoretic techniques such as cut-elimination of appropriate semiformal systems and asymmetrical interpretations in standard structures for explicit mathematics. Download Export citation Bookmark 10 citations
1995 European Summer Meeting of the Association for Symbolic Logic.Johann A. Makowsky - 1997 - Bulletin of Symbolic Logic 3 (1):73-147.details Download Export citation Bookmark
The non-constructive μ operator, fixed point theories with ordinals, and the bar rule.Thomas Strahm - 2000 - Annals of Pure and Applied Logic 104 (1-3):305-324.details This paper deals with the proof theory of first-order applicative theories with non-constructive μ operator and a form of the bar rule, yielding systems of ordinal strength Γ0 and 20, respectively. Relevant use is made of fixed-point theories with ordinals plus bar rule. Download Export citation Bookmark 2 citations
The universal set and diagonalization in Frege structures.Reinhard Kahle - 2011 - Review of Symbolic Logic 4 (2):205-218.details In this paper we summarize some results about sets in Frege structures. The resulting set theory is discussed with respect to its historical and philosophical significance. This includes the treatment of diagonalization in the presence of a universal set. Download Export citation Bookmark 2 citations
Notes on some second-order systems of iterated inductive definitions and Π 1 1 -comprehensions and relevant subsystems of set theory. [REVIEW]Kentaro Fujimoto - 2015 - Annals of Pure and Applied Logic 166 (4):409-463.details Download Export citation Bookmark 2 citations
Universes over Frege structures.Reinhard Kahle - 2003 - Annals of Pure and Applied Logic 119 (1-3):191-223.details In this paper, we study a concept of universe for a truth predicate over applicative theories. A proof-theoretic analysis is given by use of transfinitely iterated fixed point theories . The lower bound is obtained by a syntactical interpretation of these theories. Thus, universes over Frege structures represent a syntactically expressive framework of metapredicative theories in the context of applicative theories. Download Export citation Bookmark 5 citations

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