Set theory, as developed within the tradition of Cantor and Zermelo, is a mathematical discipline that is autonomous relative to logic.
According to the so-called iterative, or cumulative, conception of set, sets are mathematical objects built up from previously given objects by a process of “collecting them together”. More specifically, sets are formed in a transfinite succession of stages. At stage 0, the empty set is formed together with sets that only contain objects that are not sets (individuals). The sets formed at a successor stage a+1 are all possible collections of objects (individuals and sets) that are available at stage a. At limit stages all objects are collected together that have been obtained previously. The set-theoretic universe V contains all the objects (sets and individuals) that have been obtained in this process. The universe itself as well as such “inconsistent” totalities as the totality of ordinals, cardinals, and the totality of all sets that are not members of themselves, do not form sets. In this way the usual paradoxes of naive set theory are avoided.
Frege, on the other hand, thought of sets—or classes—as “logical objects” which are definable within his logical system as extensions of concepts. Frege’s system is a higher-order logic, where the individual variables are taken to be absolutely unrestricted, ranging over absolutely all objects, and higher-order variables are interpreted as ranging over “unsaturated” entities—Fregean functions and concepts. Frege assumes that every concept of objects determines an object, namely, the class of all objects that fall under the concept. This assumption, in conjunction with strong existence assumptions concerning concepts, implies that Frege’s foundational system is inconsistent.
In this talk I am going to discuss various proposals for developing a set theory along Fregean lines. In particular, I am going to consider various proposals for restricting Frege’s unlimited set-comprehension axiom.
Föredrag vid Filosofidagarna 2007, 8-10 juni 2007. Filosofidagarna är en nationell filosofikonferens som sedan 1995 vartannat år anordnas av Svenska Filosofisällskapet. Värdskapet för konferensen växlar mellan olika filosofiinstitutioner.