 Given these preparations, Dedekind introduces the natural numbers as follows: First, he proves that every infinite set contains a simply infinite subset. Then he establishes that any two simply infinite systems (any two models of the Dedekind-Peano axioms) are isomorphic (so that the axiom system is categorical). Third, he observes that, as a consequence, exactly the same arithmetic truths hold for all simple infinities; or closer to Dedekind’s actual way of stating this point, any truth about one of them can be translated, via the isomorphism, into a truth about the other. (That is to say, all models of the Dedekind-Peano axioms are “logically  equivalent”, which means that the axiom system is “semantically  complete”; cf. Awodey &amp; Reck 2002). In those respects,  each simply infinity is as good as any other.