Theory interpreted in non-set domain of discourse may be consistent?

Question

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:

1. predefined single object E

2. objects which may be whatever You
   like to name by letters from finite alphabet ( not necessary elements of defined set)

3. 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:

1. Is theory T first order theory? When interpreted on set domain, it is. But if not in set domain?

2. Is it necessary to interpret it in set universum, by means of any other arguments than arising from question 1*?

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

4. If we drop requirements that universum of objects is in set, will this theory becomes inconsistent?

5. 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 "the 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

Answer 1

It appears that you yearn to study various first-order theories, but do not want to be constrained by any requirement that your models, or domains of discourse, be sets. There are several ways to take such a proposal.

On the one hand, many mathematicians have yearnings similar to yours, and this has led them to try to use category theory as a theoretical background for their mathematical investigations. Surely this is part of the attraction of category theory, and some promote category theory as a kind of alternative foundation of mathematics (that is, alternative to set theory) for precisely this kind of reason. But you say that you are not especially interested in adopting that view.

Another way to study mathematical structures that are not sets, while keeping a principally set-theoretic background, is to focus on the set-class distinction in set theory. If V is the universe of all sets, we can define certain classes in V, such as { x | φ(x) }, where φ is any property. Such a class is not always a set. For example, the Russell paradox is based on the observations that if the collection R = { x | x ∉ x } were a set, then R ∈ R iff R ∉ R, a contradiction. So R is not a set. But R is still a collection of sorts, and we call it a class. A proper class is a class that is not a set. In ZFC, one can treat classes and proper classes by manipulating their definitions. That is, the classes do not exist as objects within the set-theoretic universe, but rather as definable subcollections of the universe. The intuition is that proper classes are simply too big to be sets. Other proper classes would include the class V itself (consisting of all sets), the class of all ordinals, the class of all cardinals, the class of all groups, all rings, all monoids, etc. Each of these classes is too large to be a set, but each has a perfectly clear definition defining a family of objects.

There are other formalizations of set theory, such as Goedel-Bernays set theory GBC and Kelly-Morse set theory, that allow one to treat classes as objects. In these theories, there are two kinds of objects, sets and classes, and every set is a class, but there are classes that are not sets (such as those I listed above). It turns out that GBC is a conservative extension of ZFC, which means that the assertions purely about sets that are provable in GBC are exactly the same as the assertions about sets that are provable in ZFC. Kelly-Morse, in contrast, is not conservative over ZFC, and it implies, in particular, that their must be set models of ZFC.

Now, the point is that you could study magmas that are proper classes. These would not be sets, but would still exist and could be formally analyzed as mathematical structures in these various set theoretic backgrounds. For example, one magma is simply the set union operation: (a,b) maps to (a U b), defined on all pairs of sets a, b. This magma is not a set, simply because it is much too large. There are innumerably many other such examples.

Answer 2

Just to respond to one tiny part of your question, I belong to a group of people who believe that traditional first-order logic's requirement that all domains be nonempty is misguided. This becomes of particular importance when doing constructive mathematics, where emptiness or nonemptiness may not be a decidable property. But even with classical logic, I think that in general, mathematics teaches us that it's a mistake to make the empty set a special case; usually the "correct" phrasing of a definition includes the empty set. One has to take a little more care when defining logic over domains which might be empty, but I think this care is justified, since it's very important that some theories have empty models (such as the theory of magmas).