On wiki page on Fraïssé limit, it says that neither $⟨\Bbb{N}, < ⟩$ nor $⟨\Bbb{Z}, < ⟩$ are the Fraïssé limit of FCh (Fraïssé class) because although both of them are countable and have FCh as their age (the class of all finitely generated substructures), neither one is homogeneous.
Then it gives such examples as substructures $⟨ { 1 , 3 } , < ⟩$ and $⟨ { 5 , 6 } , < ⟩$, and the isomorphism $1 ↦ 5, 3 ↦ 6$ between them. It concludes that this cannot be extended to an automorphism of $⟨\Bbb{N}, < ⟩$ or $⟨\Bbb{Z}, < ⟩$, since there is no element to which we could map $2$, while still preserving the order.
I think the substructure $⟨ { 1 , 3 } , < ⟩$ and $⟨ { 5 , 6 } , < ⟩$ are sums of $a+3b$ and $5a+6b$. So the isomorphism is $\phi(a+3b)=5\phi(a)+6\phi(b)$. But if we take $a=-1, b=1$ and $a'=-2,b'=2$, then $\phi(2)=2$. So what does it mean really?
Best Answer
The last paragraph makes no sense at all to me. How do you propose to read $\phi(a+3b) = 5a'+6b'$ as the definition of a function $\mathbb{Z}\to\mathbb{Z}$?
In any case, if $\phi\colon \mathbb{Z}\to \mathbb{Z}$ is a map with the property that $\phi(1) = 5$, $\phi(3) = 6$, and $\phi(2) = 2$, then $\phi$ is not an isomorphism, since $1 < 2$, but $\phi(1) = 5 > 2 = \phi(2)$, so $\phi$ does not preserve $<$.
In fact, for any integer $n$, if $\phi\colon \mathbb{Z}\to \mathbb{Z}$ is a map with the property that $\phi(1) = 5$, $\phi(3) = 6$, and $\phi(2) = n$, then $\phi$ is not an isomorphism. This is because $1 < 2 < 3$, so for $\phi$ to be an isomorphism, we'd have to have $5 = \phi(1) < \phi(2) < \phi(3) = 6$, but there is no integer strictly between $5$ and $6$.
Edit: Upon rereading, I've realized what you mean by "sums of $a + 3b$". In the structure $(\mathbb{Z};0,+)$, the substructure generated by $1$ and $3$ would be $\{a+3b\mid a,b\in \mathbb{N}\}$ (which is just $\mathbb{N}$). But note that $+$ is not in the language! The only symbol in the language is $<$, and the substructure generated by $1$ and $3$ is just $\{1,3\}$. The isomorphism $(\{1,3\},<)\to (\{5,6\},<)$ is exactly as stated: $1\mapsto 5$, $3\mapsto 6$.