In the large Wilfrid Hodges book: Model Theory I have 2 easy questions,first on the page 164, example 2, $V$ is exactly $A$: they begin with Let $V$ be an infinite vector space (in my opinion this should read "Let $A$ be an infinite vector space") but then they say something about $A$: Thus again $A$ is minimal.
Next, what should I take for $D$ in 6.4.1. in the proof of the fact 4.5.1 ? What corresponds to $A,B$ and $C$ from the proof of 4.5.1.(the first snippet) to in 6.4.1. (the second snippet) ?
Please see 2 snippets below.
*** $\mathbf{second\ snippet}$ ***
Suppose $C$ is a counterexample (not $B$!). Say, $c$ is a tuple of parameters in $C$ and $\delta(x,c)$ is a formula such that for every $n$
$$C\models\exists^{\ge n} x \ \ \psi(x)\wedge\delta(x,c)$$
$$C\models\exists^{\ge n} x \ \ \psi(x)\wedge\neg\delta(x,c)$$
Let $B=A$, $D\succeq B$, and $g:C\to D$ be as in Theorem 6.4.1. Then
$$D\models\exists^{\ge n} x \ \ \psi(x)\wedge\delta(x,gc)$$
$$D\models\exists^{\ge n} x \ \ \psi(x)\wedge\neg\delta(x,gc)$$
This proves (a)$\rightarrow$(b) of Fact 4.5.1.
(I insist that proving Fact 4.5.1 by elementary amalgamation is insane.)