Kleene, Stephan Cole
Introduction to Metamathamatics (1952)
Third Reprint 1962 p. 65

Let us recapitulate. In the full pictute, there will be three seperate 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.

Here (b), which is formal, is not as theory in the common sense. but a system of symbols and of the objects built from symbols (described in (c)), which however forms a kind of conventional image or model for (a). On the other hand, (a) and (c), which are informal, do not have an exactly determined structure, as does (b).

Then (c) is a theory with (b) as its subject matter, which must apply to (b) without looking at (a), or more properly without looking at the interprtation of (b) in terms of (a).

Furthermore (c) is restricted to the use of finitary methods, while in general (a) will not be.