 Starting with the classical real line \(\Re\), a set-theoretic universe—the standard universe—is first constructed over it: here by such a universe is meant a set \(U\) containing \(\Re\) which is closed under the usual set-theoretic operations of union, power set, Cartesian products and subsets. Now write \(\fU\) for the structure \((U, \in)\), where \(\in\) is the usual membership relation on \(U\): associated with this is the extension \(\cL(U)\) of the first-order language of set theory to include a name \(\boldsymbol{u}\) for each element \(u\) of \(U\). Now, using the well-known compactness theorem for first-order logic, \(\fU\) is extended to a new structure \(^*\fU = (^*U\), \(^*{\in})\), called a nonstandard universe, satisfying the following key principle: