I've missed something in my explanation of models. Allow me to define them more precisely.

Intuitively, we want an "interpretation" of the sentences generated by a logic+language which assigns each to a truth value. Model theory formalizes this as an object+relation, but we can also look at it as a map from sentences onto Z_2.

Any such means of assigning a sentence to {true, false} (or equivalent) is an "interpretation" of sorts, but not necessarily a model. We reserve the term "model" specifically for not-stupid interpretations (ones where the interpretation maps "x" and "not x" to different values, etc.)

In your construction, when you consider a surjection from sentences to some set and pick one element of the range to be "truth", you've essentially defined an interpretation in a roundabout way. (Every sentence mapped to 1 is true in that interpretation, every sentence mapped elsewhere is false in that interpretation.)

If your interpretation obeys the rules of logic ("x*y" maps to 1 whenever "x" maps to 1 and "y" maps to 1) then it's a model. Otherwise, model theory doesn't have much to say about it.

I'm not sure I understand the construct you're describing: does the above help at all? I'm not sure if I'm answering the right questions.

Don't worry about decidability in this context, I think it might be confusing things somewhat. The point I was making earlier about completeness is this:

If you consider the set of all sentences as the domain of your map, then a "theory" T is just a subset of the domain of your map. If there are multiple models (functions which obey the rules of the logic) from the set of all sentences onto Z_2 which map the subset of all sentences (theory) T to 1, then T is "incomplete".