ok. some rant about definition and semantics. integers are isomorphic ordered pairs of natural numbers (a,b) w/ equivalence relation (a,b)=(c,d) iff a+d=b+c. reals are convergent sequences of rationals, etc. in mathematics, are integers simply isomorphic to the ordered pairs of natural numbers w/ the equivalence relation? or is the set of integers equal to the set of pairs of natural numbers with the equivalence relation? is it really correct to say that "the set of naturals is a subset of the set of reals" or "reals are a subset of complexes"?