I am reading Enderton's Logic Book and I can't understand the proof that there is a non-standard arithmetic model. The proof is typical using the compactness theorem:
First we expand $\mathcal{L}$ with the constant $c$ and then we consider $\Sigma=\{0<c,\text{S}0<c,\text{SS}0<c,\cdots\}.$
Then by the compactness theorem it is possible to prove that $\text{Th}(\mathfrak{N})\cup\Sigma$ has a model $\mathfrak{M}$.
Finally we "ignore" the interpretation of the constant and then we have a model $\mathfrak{M}_0$ elementarily equivalent to $\mathfrak{N}$. We can also show that $\mathfrak{M}_0$ and $\mathfrak{N}$ are not isomorphic because the universe of $\mathfrak{M}_0$ contains the infinite number $c^\mathfrak{M}$.
What I don't understand is that if there is a non-standard element in $\mathfrak{M}_0$ it would be true that:
$\mathfrak{N}\nvDash(\exists x)(\forall y)(y<x)$
$\mathfrak{M}_0\vDash(\exists x)(\forall y)(y<x)$ Since $c^\mathfrak{M}$ would be a witness.
And the structures would not be elementarly equivalent.
$c^{\mathfrak{M}}$ is not a greatest element of $\mathfrak{M}$. For example, $S(c^{\mathfrak{M}})$ is larger than it.