Completeness and Categoricity: Frege, Godel and Model Theory

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 Godel'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 Godel's completeness theorem of 1930 and the subsequent development of first- and higher-order model theory.