Do metatheory and metalanguage themselves also form a formal system?
Do they axiomazitize formal systems, by viewing formal systems as models?
Metatheories seem more abstract than formal systems, why are metatheories said to be intuitive rather than formal, compared to formal systems, as from this link:
A metatheory exists outside the formalized object theory—the meaningless symbols and relations and (well-formed-) strings of symbols. The metatheory comments on (describes, interprets, illustrates) these meaningless objects using "intuitive" notions and "ordinary language". Like the object theory, the metatheory should be disciplined, perhaps even quasi-formal itself, but in general the interpretations of objects and rules are intuitive rather than formal. Kleene requires that the methods of a metatheory (at least for the purposes of metamathematics) be finite, conceivable, and performable; these methods cannot appeal to the completed infinite. "Proofs of existence shall give, at least implicitly, a method for constructing the object which is being proved to exist."3 (p. 64)
Kleene summarizes this as follows: "In the full picture there will be three separate and distinct "theories""
- "(a) the informal theory of which the formal system constitutes a formalization
- "(b) the formal system or object theory, and
- "(c) the metatheory, in which the formal system is described and studied" (p. 65)
He goes on to say that object theory (b) is not a "theory" in the conventional sense, but rather is "a system of symbols and of objects built from symbols (described from (c))".
Is metamathematics a special example of metatheory, when metatheory is applied to mathematical formal systems which are particular examples of formal system?
Does metamathematics mean the same as foundations of mathematics? What does metamathematics include, for example set theory, category theory, and/or logic?
Or does logic (system) exist beyond mathematics, i.e. does logic (system) not belong to mathematics or is it not seen as a branch of mathematics?
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