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On extremal axioms

Rudolf Carnap, Friedrich Bachmann & H. G. Bohnert

History and Philosophy of Logic 2 (1-2):67-85 (1981)

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Carnap’s early metatheory: scope and limits.Georg Schiemer, Richard Zach & Erich Reck - 2017 - Synthese 194 (1):33-65.details In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to (...) Download Export citation Bookmark 4 citations
Uniting model theory and the universalist tradition of logic: Carnap’s early axiomatics.Iris Loeb - 2014 - Synthese 191 (12):2815-2833.details We shift attention from the development of model theory for demarcated languages to the development of this theory for fragments of a language. Although it is often assumed that model theory for demarcated languages is not compatible with a universalist conception of logic, no one has denied that model theory for fragments of a language can be compatible with that conception. It thus seems unwarranted to ignore the universalist tradition in the search for the origins and development of model theory. (...) Download Export citation Bookmark 4 citations
(1 other version)Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (1):1-30.details This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) Download Export citation Bookmark 53 citations
Completeness and categoricty, part II: 20th century metalogic to 21st century semantics.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (1):77-92.details This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) Download Export citation Bookmark 22 citations
(1 other version)Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (1):1-30.details Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics. Download Export citation Bookmark 27 citations
From completeness to archimedean completenes.Philip Ehrlich - 1997 - Synthese 110 (1):57-76.details Download Export citation Bookmark 14 citations
Carnap’s Early Semantics.Georg Schiemer - 2013 - Erkenntnis 78 (3):487-522.details This paper concerns Carnap’s early contributions to formal semantics in his work on general axiomatics between 1928 and 1936. Its main focus is on whether he held a variable domain conception of models. I argue that interpreting Carnap’s account in terms of a fixed domain approach fails to describe his premodern understanding of formal models. By drawing attention to the second part of Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik, an alternative interpretation of the notions ‘model’, ‘model extension’ and ‘submodel’ (...) Download Export citation Bookmark 14 citations
A Note on Intended and Standard Models.Jerzy Pogonowski - 2020 - Studia Humana 9 (3-4):131-139.details This note discusses some problems concerning intended, standard, and nonstandard models of mathematical theories. We pay attention to the role of extremal axioms in attempts at a unique characterization of the intended models. We recall also Jan Woleński’s views on these issues. Download Export citation Bookmark 1 citation
Submodels in Carnap’s Early Axiomatics Revisited.Iris Loeb - 2014 - Erkenntnis 79 (2):405-429.details G. Schiemer has recently ascribed to Carnap the so-called domains-as-fields conception of models, which he subsequently used to defend Carnap’s treatment of extremal axioms against J. Hintikka’s criticism that the number of tuples in a relation, and not the domain of discourse, is optimised in Carnap’s treatment. We will argue by a careful textual analysis, however, that this domains-as-fields conception cannot be applied to Carnap’s early semantics, because it includes a notion of submodel and subrelation that is not only absent (...) Download Export citation Bookmark 3 citations
Pasch's empiricism as methodological structuralism.Dirk Schlimm - 2020 - In Erich H. Reck & Georg Schiemer (eds.), The Pre-History of Mathematical Structuralism. Oxford: Oxford University Press. pp. 80-105.details Download Export citation Bookmark 1 citation

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