Logic and sets

Abstract

The notion of the extension of a concept has been used in logic for a long time. It is usually considered to be closely connected to the intuitive notion of a set and thus seems as though it should be embedded into set theory. However, there are significant differences between this “logical” concept of set and the notion of set (class) as defined via standard axiomatic systems of set theory; it may, therefore, be quite misleading to consider the two concepts as being continuous with each other. When we look at the writings of Gottlob Frege and consider the development of his attitude to extensions, we can see what the differences consist in and which of the two notions is more apt to be used in foundations of logic. Frege himself eventually rejected sets entirely.