Informal Theory, Object Theory, and Metatheory
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." (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))".
Read more about this topic: Object Theory
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