Preliminaries

Abstract

In Cantor’s naive set theory every collection of objects specified by some property was called a set. As well-known this approach leads to contradictions, e.g. to the Russell antinomy of the set R of all sets not members of themselves (we obtain

$$ [R \in R] \Leftrightarrow [R \notin R] $$

provided R is a set). In order to block this contradiction we introduce two types of collections: classes and sets. Then a class is a collection of objects specified by some property, whereas a set is a class which is a member of some class. Thus R is no set but a (proper) class and also the concept of the class of all sets makes sense. The axiomatic set theory tries to avoid further antinomies. The axiomatic approach of Gödel, Bernays and von Neumann is suitable to handle classes and sets. For further details the interested reader is referred to Dugundji [25] although it is not necessary for understanding this book to know all the details.