David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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History and Philosophy of Logic 18 (2):79-93 (1997)
Frege?s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel?s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ?complete? it is clear from Dedekind?s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical or complete, there are logical extensions of these theories into second-order and by the addition of generalized quantifiers which are categorical. Frege?s project really found success through Gödel?s completeness theorem of 1930 and the subsequent development of first- and higher-order model theory
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References found in this work BETA
Warren D. Goldfarb (1979). Logic in the Twenties: The Nature of the Quantifier. Journal of Symbolic Logic 44 (3):351-368.
Alfred Tarski (1968). Undecidable Theories. Amsterdam, North-Holland Pub. Co..
Abraham Robinson (1963). Introduction to Model Theory and to the Metamathematics of Algebra. North-Holland.
Citations of this work BETA
Steve Awodey & Erich H. Reck (2002). Completeness and Categoricity. Part I: Nineteenth-Century Axiomatics to Twentieth-Century Metalogic. History and Philosophy of Logic 23 (1):1-30.
Steve Awodey & Erich H. Reck (2002). Completeness and Categoricity, Part II: Twentieth-Century Metalogic to Twenty-First-Century Semantics. History and Philosophy of Logic 23 (2):77-94.
Catarina Dutilh Novaes (forthcoming). Axiomatizations of Arithmetic and the First-Order/Second-Order Divide. Synthese.
Toby Meadows (2013). WHAT CAN A CATEGORICITY THEOREM TELL US? Review of Symbolic Logic (3):524-544.
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