Set theory is a branch of modern mathematics with a unique place because various other branches can be formally defined within it (Suppes 1972). For example, Book 1 of the influential works of N. Bourbaki is devoted to the theory of sets, which provides the framework for the whole enterprise. Bourbaki has said in 1949: ``... all mathematical theories may be regarded as extensions of the general theory of sets ... on these foundations I can state that I can build up the whole of the mathematics of the present day'' (Goldblatt 1984). Indeed, one can represent a natural number as a set, a rational number as a pair of natural numbers, a real number as a set of rationals, and so on (Mac Lane 1986). Hence, most of the mathematical entities may be regarded as sets and set theory can be considered as the fundamental theory underlying mathematics (cf. Note 1).
This brings up the possibility of using set theory in foundational studies in artificial intelligence (AI), particularly in commonsense reasoning. McCarthy (1983) has emphasized the need for foundational research in AI and claimed that AI needs mathematical and logical theory involving conceptual innovations. He stated that one of the key problems is the formalization of commonsense knowledge and reasoning. In his opening address in IJCAI--85 (McCarthy 1985), he stressed the feasibility of using set theory in AI and invited researchers to concentrate on the subject.
There is, in some sense, great beauty, economy, and naturalness in using sets for modeling and knowledge representation in AI (Akman 1992). This is owing to the fact that sets agree very well with our intuitions (Parsons 1990). Gödel (1947) eloquently states this in the following excerpt:
``But despite the remoteness from sense-experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense-perception.''
In this survey, we first give a brief review of classical set theories, trying to avoid the technical details---which the reader can find in classical texts like (Halmos 1974) or (Fraenkel et al. 1973)---and instead focusing on the underlying concepts. We then consider the alternative set theories which have been proposed throughout the century to overcome the limitations of classical theories. Later, we investigate the properties of a possible commonsense set theory, treating different aspects such as urelements, cumulative hierarchy, self-reference, cardinality, well-orderings, and so on. We finally summarize the noteworthy research on the subject and offer our concluding remarks.