The proof of existence of Fraissè limit as done in Hodges is based on the following claim:
Here $K$ is the family of which we are taking the limit, so it is a countable familiy of finitely generated structures which has the hereditary property, the joint embedding property and the amalgamation property.
Assuming the claim, the fact is clear.
What does not convince me is the proof of the claim, which is based on the construction of set of pairs of structures as follows:
I really can't see why this choice of $D_{k+1}$ produces the desired property. Any tips?
Thanks in advance.
To show that the given construction verifies the claim (1.8), suppose we're given $A\subseteq B$ in $\mathbf K$ and $f:A\to D_i$ as in (1.8). We need to find a suitable $j$ and an embedding $B\to D_j$ that extends $f$. Well, once $D_i$ has been constructed, the triple $(f,A,B)$ will be one of the $(f_{i,j},A_{i,j},B_{i,j})$ listed at that stage. Fix such a $j$, and let $q=\pi(i,j)$. Then at stage $q$ the construction produces $D_{q+1}$ by applying amalgamation to the embeddings $A\to B$ and $f:A\to D_i$. The amalgamation provides an embedding $B\to D_{q+1}$ that extends $f$, as required in (1.8).