Metatheorem

1. A theory concerned with the investigation, analysis and description of theory itself. Theorems on objects of formalized theories in any given logical-mathematical calculus or formal system. Some of the best known metatheorems are those of Gödel's of incompleteness for formal arithmetic systems and of completeness for the predicate calculus; and in the same calculus, that of Church on the unsolubility of decision problems.

2. Theorems regarding theorems of informal mathematical theories (e.g. duality principles in algebra).

3. The theoretical proof of another theory which may have as its object one or more axioms, theorems, proofs, rules of inference, definitions or concepts. The metatheorem is the theory of the object theory. Two forms of the metatheorem exist; the metamathematical employing finitary methods and the set-theoretic predicate logic particularly as applied to all the results on the categoricalness of different axiomatic systems.

4. A meta-level theory may be used as a means of integrating the constructs of different theories. Suppose there is a theory, S, containing the sentences b, c, d, and so on; and suppose there is another theory T, containing the sentences q, r, s, and so on. Now if S and T are isomorphic, S may be termed an interpretation of T, and T an interpretation of S. In other words, there is some relation between sentences in S which is ordinally similar to a relation between the sentences in T, and there is one-to-one correspondence between the two sets. If b corresponds to q and so forth, then b is an interpretation of q in S and q is the interpretation of b in T.

5. Now the various modes of organizing and comprehending human experience each constitute a theory, in a broad sense of the term. Each is based on sensing, which it refers to an interconnected set of constructs, to which response is made, which thus affects the environment that relevant sensing is afforded. Contextual differences between theories are due to the suitability and adequacy of the respective sets of constructs as regards the analysis and interpretation of the various segments of experience. With due regard for these differences, the theories nevertheless manifest common fundamental characteristics: they interpret experiences by means of constructs, give rise to coordinate responses and find their confirmation in the manipulated environment. Thus specifically differentiated but isomorphically structured theories are obtained, wherein there is a general ordinal similarity between the propositions. The relationship between the key terms, invariant in all theories, determines isomorphism and therefore determines the absolute meaning of the propositions in a meta-theory. Thus, it becomes theoretically possible to formulate a meta-theory integrating the diverse facets of human experience.