Take the 2-minute tour ×  Right now I am reading a proof of
Downward Löwenheim-Skolem theorem in Hodges, but I am slightly
confused about a proof Hodges makes. Let me write down some of
the definitions.  Definition: Let $T$ be a first-order theory in
a first-order language $\scr{L}$. >Then a skolemisation of $T$
is a theory $T^+\supseteq T$ in a first order >language
$\scr{L}^+\supseteq \scr{L}$ such that    1. Any model of $T$
can be expanded to a model of $T^+$.   2. For every formula
$\phi(\bar{x}, y)$ of $\scr{L}^+$ with $\bar{x}$ a nonempty
tuple, there is a term $t$ of $\scr{L}^+$ such that $$ T^+\vdash
\forall \bar{x}(\exists y\phi(\bar{x}, y)\to \phi(\bar{x},
t(\bar{x}))). $$  A theory $T$ is said to be a Skolem Theory or
to have Skolem functions if $T^+:=T$ and $\scr{L}^+ = \scr{L}$
is a skolemisation of $T$, i.e. $T$ is a skolemisation of
itself.  Hodges goes on to say that for a Skolem theory $T$, and
$A$ a model of $T$ and $X$ a subset of $A$, if the skolem hull
$B:=\langle X\rangle_A$ is nonempty