Frege's Double-Correlation Thesis and Quine's Set Theories NF and ML

Abstract

There are two fundamentally different notions of a class, which, following tradition, we might call the mathematical and the logical notions, respectively. The logical notion is essentially the notion of a class as the extension of a concept, and, following Frege, we will assume that a class in this sense "simply has its being in the concept, not in the objects which belong to it" ([8], p. 183) - regardless of whether or not concepts themselves differ, as Frege assumed, "only so far as their extensions are different" (ibid., p. 118). The mathematical notion of a class, on the other hand, is essentially the notion of a class as composed of its members, i.e., of a class that has its being in the objects that belong to it. This notion of a class, we claim, is none other than the iterative concept of set - or at least that is what it comes to upon analysis. Note that although what accounts for the being of a class under the one notion is not the same as what accounts for the being of a class under the other, nevertheless the axiom of extensionality applies equally to both notions. This means that the axiom of extensionality does not of itself account for the being of a class.