Construction of a Henkin Theory I'm trying to understand Henkin's proof of Gödel's completeness theorem, specifically the construction of a Henkin theory T' with language L' from an arbitrary theory T over a language L. 
My problem with the proof is that I don't understand why does it suffice to consider the subset of all the L'-formulas with at most one free variable when extending the theory T. Isn't the Henkin property a property of all L'-formulas, even those with more than one free variable?
Thank you very much for your help.
 A: The definition of a Henkin Theory $T$ requires that for each sentence $\exists x \phi$ in the language of $T$ there is a constant $c_{\phi}$ such that $T \vdash (\exists x \phi) \Rightarrow \phi[c/x]$. I.e., $\phi$ is expected to have only the variable $x$ free. It wouldn't work to extend this to formulas $\phi$ with other free variables. E.g., consider the following formula in the language of arithmetic:
$$\phi(x, y) \mathrel{:=} \exists x [(y = 0 \Rightarrow x = 0) \land (y > 0 \Rightarrow x = 1)]$$
It would be inconsistent with the theory of arithmetic to introduce a constant $c$ such that $\phi(c, y)$ holds for all $y$.
A: Let's take a formula $\phi(x, y)$ such that the sentence $\exists x, y\phi(x, y)$ is in $T'$. Then:


*

*Consider the sentence "$\exists x(\exists y\phi(x, y))$". Our expanded language $L'$ contains a new constant $c$ for this sentence, and our expanded theory contains the sentence "$\exists y(\phi(c, y)$." 

*But now that's itself a sentence of the form "$\exists z[stuff]$." So we add a constant $d$ for it, too, and the sentence "$\phi(c, d)$."
So two-variable formulas are handled in this "two step" process; similarly for $n$-variable formulas.
