From Model Theory article from wikipedia : "A theory is satisfiable if it has a model $ M\models T$ i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set $T$". In structure definition there is also requirement for "container of a structure" to be set.
As we assume then, every model have to lay inside of some set-container. This obviously give us in serious trouble, as for set theory, there is no model of this type, and even maybe cannot be. One of possible explanations why set theory cannot be closed inside set ( which will lead us to some well known paradoxes) is opinion that "there can be no end to the process of set formation" so we have "structure" which cannot be closed inside itself which is obviously rather well state.
As we know that not every theory may have a model (see set-theory) then some question arises:
What are the coses (other than pure practical - if they are set we know how to work with them) of putting so strong requirements for model to be set?
Is there any way to weaken this requirement?
Are there any "explorations" of possible extension of model theory with fundamental objects other than sets?
I presume that some categorical point of view may be useful here, but is there any? I am aware about questions asked before, specially here 8731, but it was asked in context of category theory which is of course valid point of view but somehow too fine. I would like to ask in general.
And last one, philosophical question: is then justified, that condition for a theory to have model in set universe is some kind of anthropomorphic point of view - just because we cannot build any other structures in effective way we build what is accessible for us way but it has no objective nor universal meaning? Is true that model theory is only a "universal algebra+logic" in universe of the set, or it justifications may be extended to some broader point of view? If yes: which one?
I have hope that this question is good enough for mathoverflow: at least please try to deal wit as kindly request for references.
Remark: Well formulated point from n-CathegoryCafe discussion: "In the centre of Model Theory there is " fundamental existence theorem says that the syntactic analysis of a theory [the existence or non-existence of a contradiction] is equivalent to the semantic analysis of a theory [the existence or non-existence of a model]."
In fact the most important point is: may it be extended on non "set container" universes?
I would like to thank everyone who put here some comments or answers. In is the most interesting that in a light of answer of Joel David Hamkins it seems that for first order theories (FOT) consistency is equivalent to having set model. It is nontrivial, because it is no matter of somehow arbitrary definition of "having model" but it is related to constructive proof of Completeness Theorem of Gödel. From ontological point of view it then states that for FOT there is no weaker type of consistency than arising from model theory based on sets, and in some way it is maximal form of consistency simultaneously. Then there is no way to extend for FOT equivalence to non-set containers, which is nontrivial part - the only theories which are consistent in FOT are those which has a set models and this statement is proved not using set theoretical constructions in nonconstructive ways. So it was important to me, and I learn a lot from this even if for specialist it is somehow maybe obvious. I have hope that I understand it right;-)
@Tran Chieu Minh: thank You for pointing to interesting discussion, I will try to understand the meaning of Your remarks here and there.