A proof of predicate logic inability to express graph reachibility (page 63) involves a formula which can be interpreted as (there is no path, no matter what is the length) or (there is some path). The proof says that it is "unsatifiable by construction," which means means that some example is provided. Which example are they talking about?
Let $\Delta = \{ \neq \phi_n \mid n \geq 0 \} \cup \{ \phi [c/u][c'/v] \}$. By construction, $\Delta$ is unsatisfiable, because the first set says, "There is no path of any finite length from $c$ to $c'$, and the second set says that there is one.
I would say that $\Delta$ is not just satisfiable but it is valid (I hardly can come up with an example where it is not the case). Anyway, my question is what is this example has to do with the prove by construction? The proof by constraction assumes that you build an example of what to be (dis)proved. Which lemma is proved by (construction of) $\Delta$?
For your first question, 'unsatisfiable by construction' means that the formula itself has been constructed such that there is no way to satisfy it. A model that satisfies it would have some finite path from $c$ to $c'$ by $\phi[c/u][c'/v]$. That path has a length that is some natural number $n$. But by $\{ \neg \phi_n | n \geq 0 \}$, for all $n$ there is no path from $c$ to $c'$ of length $n$.
For your second question, I'm not sure where it says that $\{ \neg \phi_n | n \geq 0\}$ is a finite set; clearly it isn't, since it has one element for each nonnegative integer.