As an afterthought to this question on sets in set theory, and more specifically to the observation that a (first-order) logic requires a meta-language to explain itself (i.e. there is already an implication sign in the definition of rules of such a logic, which differ from the sign symbol of the logic itself), I wonder:
Are there attempts to close the problem of regression of meta languages (and then meta languages for these and so on)?
My idea would be that it might be possible to close the loop by having a structure that is able to define/model a framework, which is able to state axioms of a logic, which exactly mirror the first one. At least for some theories.
In what sense can some "structures" define a framework which is strong enouth to express a copy of the original "structures"?