I'm currently learning some introductory model theory from Marker's "Model Theory: An Introduction", Kaye's "Models of Peano Arithmetic", and Hodges' "Model Theory", and I am confused by the Wikipedia article on Peano Arithmetic. I am interested in the question I posed in the title and this article is really confusing me.

It states there that,

A model of the Peano axioms is a triple $(\mathbb{N}, 0, S)$, where $\mathbb{N}$ an infinite set, $0 \in \mathbb{N}$ and $S : \mathbb{N} \rightarrow \mathbb{N}$ satisfies the axioms. Dedekind proved in his 1888 book, What are numbers and what should they do (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 $(\mathbb{N}_A, 0_A, S_A)$ and $(\mathbb{N}_B, 0_B, S_B)$ of the Peano axioms, there is a unique homomorphism $f : \mathbb{N}_A \rightarrow \mathbb{N}_B$ satisfying, $$f(0_A)=0_B$$ and $$f(S_{A}(n))=S_{B}(f(n))$$ and it is a bijection.

Doesn't Tennenbaums theorem, and the existence of non-standard models, show that not all models of peano arithmetic are isomorphic? If this is the case, then what result did Dedekind actually prove and does anyone know where I can find a reference to the theorem he proved or a proof of it?

Was sind und was sollen die Zahlenis the original source, and Dover has an English editionEssays on the Theory of Numbers(which contains that and also Dedekind's 1872 article where Dedekind cuts are introduced). $\endgroup$ – Ed Dean Mar 11 '12 at 21:13