I have been reading the Wiki articles for metatheory and metalanguage, but not sure if I have understood what they are about. Some accessible examples may help clarify a bit, I guess.

  1. Do metatheory and metalanguage themselves also form a formal system?

    Do they axiomazitize formal systems, by viewing formal systems as models?

  2. 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))".

  3. 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?

Thanks and regards!

  • $\begingroup$ The last question can't really get answered, or at least any answer given seems contentious at best. $\endgroup$ – Doug Spoonwood Feb 17 '12 at 3:15
  • $\begingroup$ @DougSpoonwood: Thanks! I mean in consistent and rigorous sense, consistent with what logic systems as formal systems are, what mathematical formal systems are, ... $\endgroup$ – Tim Feb 17 '12 at 3:20
  • $\begingroup$ I think it's still not that easy, so to speak. A formal system can qualify as unsound, inconsistent, and incomplete (among other "undesireable" metalogical properties). That an inconsistent, unsound, and incomplete formal system satisfies the definition of a formal system will not pose a problem for anyone, I believe. But whether it belongs to the study of "logic" or "mathematics" proper, I believe, will lead to an argument. $\endgroup$ – Doug Spoonwood Feb 17 '12 at 3:53

The metalanguage doesn't axiomitize a formal system. The axioms of a formal system actually exist in the object language. The metalanugage consists of a separate language than the object language to make statements about the object logic, such as how rules of inference for the formal system work, among other purposes (such as metatheorems like a deduction theorem for the theory, completeness theorem, soundness theorem, etc.). One might say that you have a metalanguage, because you can't describe how things work in the object language. As I understand it, in the object language you can only "observe" what exists there. The object language (in the context of a formal system) strictly consists of formulas permissible by the formation rules. You can't even say that a formula is true (tautology, or theorem if the logical system is sound) in the object language.

  • $\begingroup$ +1 Thanks! I wonder if object language and formal language are the same thing? $\endgroup$ – Tim Feb 17 '12 at 3:58
  • $\begingroup$ @Tim No, by which I mean "not always". An object language just consists the language under study at the moment. It consists of the "object" you're studying. That could consist of a formal language, though you could consist of something informal, like say ordinary lattice theory (insert whatever ordinary mathematical theory you're studying... calculus, group theory, linear algebra, topology, etc. UNLESS you have at least a set of formation rules present for formulas), en.wikipedia.org/wiki/Object_language. $\endgroup$ – Doug Spoonwood Feb 17 '12 at 4:20

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