Question from Hodges' textbook Shorter Model Theory

Question

I'm thinkng through Section 2.3. of Hodges' textbook Shorter Model Theory, problem 7(b):

"Let $L$ be a first-order language. Show (without assuming that every structure is non-empty ) that every formula $\phi$(x) of $L$ is equivalent to a prenex formula $\psi$(x) of $L$. [You only need a new argument when x is empty.]"

My idea is that we can Skolemize the theory $T$ to get a universal theory $T$`.Is there some reason why this would not solve the problem?

Thanks!

Note: Since posting this, I've realized Skolemization won't work because we need $\phi$, $\psi$ to be from the same language.

Answer 1

Let $\phi$ be a sentence (i.e. no free variables) and let $\psi$ be its prenex form that works for all structures except the empty structure. Now to make $\psi$ work in the empty structure as well we need to add a dummy quantifier.

Recall that if $x$ is not free in $\chi$, then $\exists x \chi$ is equivalent to $\chi$ on nonempty structures, but is false on the empty structure. Dually $\forall x \chi$ is equivalent to $\chi$ on nonempty structures and is true in the empty structure.

Now if $\phi$ is false in the empty structure, then take $\exists x \psi$, otherwise take $\forall x \psi$.