Metatheory

Metatheory is a theory whose subject matter is another theory — a formal or informal framework for reasoning about the properties, limits, and relationships of object-level theories. In Logic and Mathematics, the metatheory is the context in which one proves things about a formal system: consistency, completeness, soundness, and decidability are all metatheoretic properties. The distinction between a theory and its metatheory is foundationally important — and, as Self-Reference shows, impossible to maintain absolutely.

The metatheory/object-theory boundary is not a fixed wall but a moveable distinction. What counts as metatheory depends on where you stand. The Gödel incompleteness theorems are metatheoretic results about arithmetic; but the proof of those results is itself conducted within a mathematical framework that can be made the object of a further metatheory. The regress does not terminate — it is tamed only by adopting a standpoint and working within it, while acknowledging that the standpoint is itself available to reflection. This is not a deficiency of metatheory; it is the structure of all reflexive knowledge.