Following the blow. I will try to ask question in order to check if I well understand what was pointed. I decide to ask another question, because mathoverflow is not projected to be good environment for discussion, so it would be better to ask another question than to modify previous one.

From wikipedia, we have definition of **magma**:

"In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M × M → M. A binary operation is closed by definition, but no other axioms are imposed on the operation."

Axioms of magma structure are first order theory. So lets drop words "set" and "closed" from the definition, and try playing with theory T for which we have:

predefined single object E

objects which may be whatever You

like to name by letters from finite alphabet ( not necessary elements of defined set)binary operation defined that for every pair of objects it sends it into predefined element E.

It has interpretation on the ground of ZFC and then it has finite or infinite models which are sets - in fact it may be considered as trivial magma if we assume that objects are from set $M U \{E\}$. Whilst considered without set background it should also work, and do not be cursed by any obvious paradoxes ( because it has set-models). Then we have an example of theory which, when interpreted in domain of sets, is first order theory and is consistent / has models.

Suppose we **formally** drop requirement that domain of discourse for this theory is set interpretation. So then several questions arises:

**Is theory T first order theory?**When interpreted on set domain, it is. But if not in set domain?Is it necessary to interpret it in set universum, by means of any other arguments than arising from question 1*?

Can we say that T is consistent in domain of set theory ( whilst we do not know in other domains)?

**If we drop requirements that universum of objects is in set, will this theory becomes inconsistent?**Is for any theory obligatory to point its domain of discourse in a formal meaning, that is for pure syntactical definition? Is interpretation necessary?

**By consistent I presume definition that ""Consistent" means that a contradiction cannot be deduced from it."**

Suppose that answer on 4th question is NO, we may say that theory is still consistent even if we drop requirement that it is interpreted on domain of sets. Then we will end with theory which has set-model. From completeness theorem when interpreted on domain of set is consistent, and even if we drop certain domain interpretation we will still have consistent theory of first order with model from sets universum, but also with models outside of it. Suppose that we get our objects from some big category. I believe that it does not changes nothing, does it? Then there are theories for which we have models which are not sets ( whilst to be consistent it still have to have models which are sets!).

Is there any obvious mistake in this hand-waving of mine?

From Wikipedia article about First Order Logic I know that "*he definition above requires that the domain of discourse of any interpretation must be a nonempty set.*" but it is a remark pointing to "empty domains" and "free logic" which obviously are not in case here. The only interesting link is this: Interpretation_(model theory) but it still is related to set or empty domain interpretations. But from category theory it seems to be possible to have theories which are interpreted in larger domains that sets in consistent way.

I would like to mention other questions in this or similar area here on mathoverflow, but I would like to say that I am not interested in category theory by other means than just an example of theory which is has other domain of interpretation than set.: Is there a relationship between model theory and category theory? Categorical foundations without set theory