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Let $\LL$ be a formal system with deductive apparatus $\DD$.

A metamodel $\MM$ for $\LL$ is an interpretation for $\LL$.

Thus, it assigns meanings to all of the undefined terms of $\LL$.

Further, it specifies a means for deciding if a sentence of $\LL$ is to be considered true in $\MM$.

It is then required that all the sentences derivable by the deductive apparatus $\DD$ are true in $\MM$.

From the above specification, it follows that a metamodel $\MM$ is a conceptual device, in that it does not require to be founded on rigorous mathematical foundations.

Thus, $\MM$ need not be a set or some other rigid notion, just something that one can contemplate, denote and convey to others.

For example, even without rigid mathematical foundations it is clear to all sane people that there is a concept "set" that one can reason about.

One could then try to determine if the collection of these "sets" is a metamodel for the language of set theory.

As another example there is the notion of metacategory, which can be described as being any metamodel for the language of category theory.

Also see