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  1. Categoricity.John Corcoran - 1980 - History and Philosophy of Logic 1 (1):187-207.
    After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...)
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  • Über Extremalaxiome.Rudolf Carnap & Friedrich Bachmann - 1936 - Erkenntnis 6 (1):166-188.
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  • 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.
    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 (...)
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  • 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.
    Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.
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  • Philosophy of Mathematics.Roman Murawski & Thomas Bedürftig (eds.) - 2018 - De Gruyter.
    The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching. It looks for answers both in mathematics and in the philosophy of mathematics from their beginnings till today. The reference point of the considerations is the introducing of the reals in the 19th century that marked an epochal turn in the foundations of mathematics. In the book problems (...)
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  • 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.
    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 (...)
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  • Carnap on extremal axioms, "completeness of the models," and categoricity.Georg Schiemer - 2012 - Review of Symbolic Logic 5 (4):613-641.
    This paper provides a historically sensitive discussion of Carnaps theory will be assessed with respect to two interpretive issues. The first concerns his mathematical sources, that is, the mathematical axioms on which his extremal axioms were based. The second concerns Carnapcompleteness of the modelss different attempts to explicate the extremal properties of a theory and puts his results in context with related metamathematical research at the time.
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  • On extremal axioms.Rudolf Carnap, Friedrich Bachmann & H. G. Bohnert - 1981 - History and Philosophy of Logic 2 (1-2):67-85.
    In the paper translated here, Carnap and Bachmann shows that the apparently metalinguistic ?extremal' axioms that are added to some axiom systems to the effect that the foregoing axioms are to apply as broadly, or as narrowly, as possible may be formulated directly as proper axioms. They analyze such axioms into four fundamental types, with the help of a concept of ?complete? isomorphism.
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  • Models and reality.Hilary Putnam - 1980 - Journal of Symbolic Logic 45 (3):464-482.
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  • On the concept of categoricity.Andrzej Grzegorczyk - 1962 - Studia Logica 13 (1):39 - 66.
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