 The saturation property expresses the intuitive idea that the nonstandard universe is very rich in comparison to the standard one. Indeed, while there may exist, for each finite subcollection \(\cF\) of a given collection of properties \(\cP\), an element of \(U\) satisfying the members of \(\cF\) in \(\fU\), there may not necessarily be an element of \(U\) satisfying all the members of \(\cP\). The saturation of \(^*\fU\) guarantees the existence of an element of \(^*U\) which satisfies, in \(^*\fU\), all the members of \(\cP\). For example, suppose the set \(\bbN\) of natural numbers is a member of \(U\); for each \(n \in \bbN\) let \(P_n (x)\) be the property \(x \in \bbN \amp n \lt x\). Then clearly, while each finite subcollection of the collection \(\cP = \{P_n : n \in \bbN\}\) is satisfiable in \(\fU\), the whole collection is not. An element of \(^*U\) satisfying \(\cP\) in \(^*\fU\) will then be an “natural number” greater than every member of \(\bbN\), that is, an infinite number.