GFO Part I Basic Principles



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			Onto-Med >> Theories >> GFO Part I Basic Principles
			Overview General Formal Ontology (GFO)	Next: 14.4 Property Relations Up: 14 Ontologically Basic Relations Previous: 14.2 Set and Set-theoretical 14.3 Instantiation and Categories $\GCat(x)$ is a predicate that represents the (meta)-category of all categories. We do not consider $\GCat$ to be an instance of itself. The symbol $\Ginst$ denotes instantiation. Its second argument is always a category, the first argument can be (almost) any entity. If the second argument is a primitive category, then the first must be an individual. Individuals - in general - can be understood as urelements with respect to instantiation. Since we assume categories of arbitrary (finite) type, there can be arbitrarily long (finite) chains of iteration of the instantiation relation. Since sets have no instances (they have elements) they can be understood as another kind of urlements w.r.t. instantiation. On the other hand, categories do not have elements, but instances, hence categories are urlements with respect to the membership relation. The definable extension relation, $\Gext(x,y)$ , is a cross-categorial relation, because it connects categories with sets and is explicitly defined in the following way: $\Gext(x,y) =_{df} Set(y) \wedge \forall u (u \in y \leftrightarrow u \Ginst x)$ . We may stipulate the existence of the set of all instances of a category by the following axiom (existence axiom): $\forall x (Cat(x) \rightarrow \exists y (ext(x,y))$ . If we assume this axiom then we may define the extensionality operator for categories: $\GExt(x) = \{ y \:\vert\: y \Ginst x \}$ . Note, that the existence axiom contradicts the foundation axiom for sets, in case of existence of non-wellfounded categories. For this reason, we do not assume the foundation axiom for sets. Next: 14.4 Property Relations Up: 14 Ontologically Basic Relations Previous: 14.2 Set and Set-theoretical Robert Hoehndorf 2006-10-18

Onto-Med >> Theories >> GFO Part I Basic Principles

14.3 Instantiation and Categories