Peano axioms
A model of the Peano axioms is a triple , where N is a (necessarily infinite) set, and satisfies the axioms above. Dedekind proved in his 1888 book, "The Nature and Meaning of Numbers" (, i.e., “What are the numbers and what are they good for?”) that any two models of the Peano axioms (including the second-order induction axiom) are isomorphic. In particular, given two models and of the Peano axioms, there is a unique homomorphism satisfying