TITLE: Henkin-style completeness proofs for intuitionistic logic
QUESTION [9 upvotes]: Henkin-style completeness proofs are founded on a few basic presuppositions, such as the assumptions that the language of a logical theory must be enumerable (or at least that the axiom of choice holds), and that it must contain a (not necessarily primitive) logical connective for negation. 
One could fairly describe its general method as follows:

By (reverse) contraposition, assume that $\Gamma \nvdash p$;
Show that $\Gamma \cup \{\neg p\}$ is consistent;
Extend $\Gamma \cup \{\neg p\}$ to a maximal consistent set $\Delta$ as follows:
            \begin{align}
    \Delta_0 :=& \Gamma \cup \{\neg p\} \\
    \Delta_{n+1} :=&
     \begin{cases}
      \Delta_n \cup \{\varphi_{n+1}\} & \text{if } \Delta_n \cup \{\varphi_{n+1}\} \text{ is consistent} \\
      \Delta_n \cup \{\neg \varphi_{n+1}\} & \text{otherwise}
     \end{cases} \\
    \Delta :=& \bigcup_{n \in \mathbb{N}} \Delta_{n}.
    \end{align}
Prove that $\Delta$ is consistent, maximal and that $\Gamma \cup \{\neg p\} \subseteq \Delta$;
Construct a model $\mathcal{M}$ s.t. $\left[\!\!\left[ \varphi \right]\!\!\right]_\mathcal{M}=1$ iff $\varphi \in \Delta$;
Derive a contradiction by showing that $\left[\!\!\left[ \Gamma \right]\!\!\right]_\mathcal{M}=1$ but $\left[\!\!\left[ p \right]\!\!\right]_\mathcal{M}=0$.

However, a moment's reflection shows that the inference from (1) to (2) fails for constructive logics, or, more precisely, logics without the inference rule of double negation elimination. In such logics, a context that does not prove $p$ may prove $\neg\neg p$ (thus being consistent with $p$). Constructively, we can only obtain a weakened form of (1)$-$(2), that is,
$$ \Gamma \nvdash \neg \neg p \quad \Longrightarrow \quad \Gamma, \neg p \nvdash \bot.$$
This reveals that, as they are commonly formulated, Henkin-style proofs can only be obtained for logical theories that allow for classical (as opposed to constructive) reasoning. In other words, the logic must be able to prove the law of the excluded middle, double negation elimination etc.
I wonder if a slight modified form of Henkin-style completeness proof is still possible for intuitionistic logic? Is there any reference on the subject? It is known to be complete with respect to topological models, Kripke semantics, and Heyting algebras. Are the proofs given using another method?

REPLY [5 votes]: Henkin-style completeness proofs for intuitionistic logic are perfectly possible: instead of maximal consistent sets, you consider, for each formula $A$, maximal sets $\Gamma$ such that $\Gamma\nvdash A$; the collection of all such sets (for all $A$ together) will form a Kripke model, where the sets are ordered by inclusion ($A$ is ignored for the ordering). In this model, each point $\Gamma$ will satisfy all formulas from $\Gamma$, and it will not satisfy the formula $A$ with respect to which $\Gamma$ was maximal.
Completeness with respect to Heyting algebras is even more trivial: just as in any other algebraizable logic, you just construct the Lindenbaum–Tarski algebra. That is, for a fixed $\Gamma$, you take the algebra of formulas with operations given by the connectives, preordered by
$$A\le B\iff\Gamma\vdash A\to B,$$
and then its quotient by the kernel equivalence relation $A\le B\land B\le A$ is a Heyting algebra in which the canonical assignment gives $\Gamma$ value $1$, and all $A$ unprovable from $\Gamma$ a smaller value.
There are some more issues with predicate logic, but I’m assuming from the question that you mean propositional logic.