Abstract

The work is primarily historical and interpretive in character, focusing on the several paradoxes of definability and their meaning for the intuitive notion of mathematical definability. The principal historical figure considered is Russell, whose ramified hierarchy of types is the primary source of analyses of definability. Certain of Russell's writings from the period 1900-1906 are examined for the light they shed on the development of Russell's approach to the paradoxes generally. In particular, attention is given to the gradual development of a commitment, on the part of Russell, to type theory as a solution to the paradoxes. With regard to Russell's later work , which is the subject to Chapter 5, there is the controversial issue of the precise relation between the vicious-circle principle and the ramified theory of types. This is related to the issue of Russell's search for arguments in support of type-restrictions. It is claimed that Russell lacks convincing arguments other than vicious-circle, with the result that type-restrictions and vicious-circle are ideas standing to one another in a relation of mutual support. ;A recurring theme in this first half of the work is what I have called the absolutist conception of mathematics, which conception incorporates platonism but which goes beyond platonism to incorporate a certain absolutism with regard to language and definability. It is not claimed that mathematicians circa 1900 generally subscribed to the absolutist conception but rather only that certain of those who first responded to the paradoxes of definability in particular held something like this view. ;Chapter 6 concerns Zermelo's notion of definite property and its role in the development of axiomatic set theory. The historical discussion of subsequent characterizations of that notion by Weyl, Fraenkel, Skolem, and Zermelo himself is geared toward establishing some link between Zermelo's notion and the intuitive notion of mathematical definability