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

 
   


14.2 Set and Set-theoretical Relations

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

$\GSet(x) \wedge \GSet(y) \rightarrow (\forall u
( u \in x \leftrightarrow
u \in y)) \rightarrow x = y)$

As sets can be nested, we can consider all set-urelements that occur in a set. First, there is the least flattened set $y = \Gtrans(x)$, which extends the nested set on the first level of nesting with all class-urelements contained in any depth of nesting. That means, $y$ satisfies the conditions $x \subseteq y$, and for every $z \in y$ holds that $z \subseteq y$. Then the class $\Gsupp(x) = \{ a \:\vert\:$$a$ is a class-urelement and $a \in \Gtrans(y) \}$, called the support of $x$, contains all class-urelements of $x$ and only them. A class $x$ is said to be pure if $\Gsupp(x) = \emptyset$.

Robert Hoehndorf 2006-10-18
 
       
     
     
     

   
     
     
       
 

deutsch   imise uni-leipzig ifi dep-of-formal-concepts