Frege on Extensions of Concepts, from 1884 to 1903

Abstract

R ussell's paradox defeated Frege's attempt to demonstrate logicism-the view that the mathematics of number can be derived from the axioms of logic together with definitions in logical terms of mathematically primitive expressions. The paradox indicated that Frege's logic was inconsistent and that the notion in terms of which he tried to define the cardinal numbers was defective. The defective notion, that of the extension of a concept, has remained interesting because it is motivated by intuitions that, arguably, play an ineliminable but inadequately understood role in modern set theories.' The development of Frege's views is also historically interesting. Frege was uncertain about the crucial notion and the paradox-producing axiom from the outset. He experimented with alternatives. What can be pieced together about his reasoning suggests deep tensions in his thought on fundamental matters. This paper is purely historical. It concentrates on the period from the publication of The Foundations of Arithmetic in 1884 to the publication of The Basic Laws of Arithmetic (first volume, 1893; second, 1903). I shall trace the development during this period of Frege's notion of the extension of a concept.