Is the standard model for the language of number theory elementarily equivalent to one with a nonstandard element? On page 89 in A Friendly Introduction to Mathematical Logic, the author writes that the standard model $\mathfrak{N}$ for $\mathcal{L}_{NT}$ is elementarily equivalent to a model $\mathfrak{A}$ that has an element of the universe $c$ that is larger than all other numbers.
I'm new to mathematical logic, but I understand that elementarily equivalent means the two structures have the same set of true sentences. However, it seems to me that the following sentence is true in $\mathfrak{A}$ but not in $\mathfrak{N}$. What am I missing?
$\exists x\ \forall y\ (x=y \vee y<x)$

In the notes, I don't see the claim that $c$ is larger than all other numbers of $\mathfrak{A}$. The number $c$ in $\mathfrak{A}$ is larger than $0$, $S(0)$, $S(S(0))$, etc., - so $c$ is greater than every element of $\mathfrak{N}$. But there will be other elements of $\mathfrak{A}$ that are larger than $c$. Not every element of $\mathfrak{A}$ is of the form $S^n(0)$ for some $n \in \mathfrak{N}$. 
