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N subset 0 Z, Z subset 1 Q, Q subset 2 R, R subset 3 Cwhere the relations subset i denote the corresponding embeddings that we will discuss in Section Embedding Sets. These embeddings preserve all relevant notions (functions and predicates) and the corresponding properties such that we can operate in the domains as if we actually had the relationship
N subset Z, Z subset Q, Q subset R, R subset C.Indeed there exists for every domain D an "identical twin" D' which is a subset of the next larger domain:
We will from now on operate with these twins pretending that they are the siblings that we have actually defined.
Consequently, we do not any more bother whether + denotes +Z or +Q; they have essentially the same properties with respect to computing and reasoning. Of course, we must still take care of that some functions have no result in a particular domain (the difference of two natural numbers is not necessarily a natural number) but only in an enclosing domain (the difference of two natural numbers is always an integer number).
