### Variants of Classical Set Theory and their Applications

Professor Peter Aczel

Classical Axiomatic Set Theory, as formalised in ZFC; i.e. Zermelo Fraenkel set theory with the Axiom of Choice, has been used, through much of this century, as the foundational theory for modern pure mathematics. This central role for ZFC is based on the fact that all the mathematical objects needed can be coded in purely set theoretical terms and their properties can be proved from the ten or so axioms of ZFC.

For various reasons many other systems of set theory have been studied by logicians and others. In Manchester three kinds of variants have received particular attention. These are

• Hyperset Theory
• Generalised Set Theory
• Constructive Set Theory

#### Hyperset Theory

This is axiomatic set theory, modified by dropping the Axiom of Foundation and instead adding some of a variety of possible axioms that assert the existence of non-well-founded sets. While the Axiom of Foundation is usually thought to be true for the standard iterative-combinatorial conception of set in which sets are thought of as being `formed' out of their elements, the axiom plays very little role in the coding of mathematical objects. In the last decade or two non-well-founded sets have turned out to be useful in allowing simple ways to code various kinds of circular and non-well-founded objects. This has given rise to some elegant mathematics and the application of non-well-founded sets to linguistics, computer science and philosophy, more specifically, in linguistics to situation theory (a mathematical theory that gives a foundation for the situation semantics approach to the semantics of natural language), in computer science to process algebra and final semantics, and in philosophy to an approach to the Liar Paradox.

#### Generalised Set Theory

This is axiomatic set theory, modified to allow for other kinds of objects besides pure sets. The simplest modification is reasonably familiar. This is to allow for atoms (also called urelemente). These are objects that have no internal structure. As well as sets having sets as elements they may have atoms also as elements. But it is also possible to allow for non-sets that do have internal structure. For example axiomatic set theory can be modified so as to allow for a primitive notion of ordered pair. So given any two objects a,b of the universe, sets, atoms or whatever, as well as being allowed to form the unordered pair set {a,b}, it can be allowed to form a new object (a,b), called the ordered pair of a,b. This object is not a set but, like {a,b} is a structured object having internal components a,b. More elaborate kinds of structured objects that are not sets can also be allowed.

#### Constructive Set Theory

Instead of changing the non-logical axioms of axiomatic set theory so as to allow for non-well-founded sets or for non-sets of various kinds we may consider changing the logic. One possibility is to replace classical logic by intuitionistic logic. Provided that the non-logical axioms of axiomatic set theory are carefully formulated the resulting set theory has been called Intuitionistic Set Theory. Constructive Set Theory is intended to be a set theoretical approach to constructive mathematics. Intuitionistic Set Theory would seem to be too strong to be taken to be an axiomatic constructive set theory. Various much weaker subsystems seem to be more appropriate. In particular there has been a good deal of attention focused on an axiom system CZF.