Completeness and categoricity: Frege, gödel and model theory

History and Philosophy of Logic 18 (2):79-93 (1997)
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Abstract

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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Stephen Read
University of St. Andrews

References found in this work

Model-Theoretic Logics.Jon Barwise & Solomon Feferman - 2017 - Cambridge University Press.
Foundations of Set Theory.Abraham Adolf Fraenkel & Yehoshua Bar-Hillel - 1973 - Atlantic Highlands, NJ, USA: Elsevier.
Undecidable theories.Alfred Tarski - 1953 - Amsterdam,: North-Holland Pub. Co.. Edited by Andrzej Mostowski & Raphael M. Robinson.
From Frege to Gödel.Jean van Heijenoort - 1968 - Philosophy of Science 35 (1):72-72.

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