Peano Axioms - Models

Models

A model of the Peano axioms is a triple (N, 0, S), where N is an infinite set, 0 ∈ N and S : N → N satisfies the axioms above. Dedekind proved in his 1888 book, What are numbers and what should they be (German: Was sind und was sollen die Zahlen) that any two models of the Peano axioms (including the second-order induction axiom) are isomorphic. In particular, given two models (NA, 0A, SA) and (NB, 0B, SB) of the Peano axioms, there is a unique homomorphism f : NA → NB satisfying

\begin{align} f(0_A) &= 0_B \\ f(S_A (n)) &= S_B (f (n)) \end{align}

and it is a bijection. The second-order Peano axioms are thus categorical; this is not the case with any first-order reformulation of the Peano axioms, however.

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