GFO Part I Basic Principles
 Ontological Investigations Conceptual Modelling Onto-Builder Axiomatic Foundations Domain Ontologies Onco-Workstation Metalogical Analyses Ontology Languages SOP-Creator

14.2 Set and Set-theoretical Relations

The membership relation is the basic relation of set theory. denotes the category of all sets, represented as a unary predicate. Usually, the notation is used for type-free systems (e.g. ZF), but it may be adapted for typed languages. implies that either and are both sets, or is a so-called class-urelement and is a set. The subset relationship is defined in terms of membership: . We include in the ontology of sets an axiomatic fragment of formal set theory, say of ZF, in particular, the axiom of extensionality:

As sets can be nested, we can consider all set-urelements that occur in a set. First, there is the least flattened set , which extends the nested set on the first level of nesting with all class-urelements contained in any depth of nesting. That means, satisfies the conditions , and for every holds that . Then the class  is a class-urelement and , called the support of , contains all class-urelements of and only them. A class is said to be pure if .

Robert Hoehndorf 2006-10-18