Metatheorems

Metatheorems are theorems that concern mathematical theories themselves rather than particular statements within those theories. In logic and foundations, a metatheorem asserts something about the structure, proof systems, models, or meta-properties of one or more formal theories. They examine what can be proven, what can be modeled, or what limits exist for the theories and their proof procedures.

A metatheorem may address consistency, completeness, soundness, decidability, conservativity, and interpretability. It might state that a given deductive system is sound with respect to a class of models, that every satisfiable set of sentences has a model, that a theory conservatively extends another, or that one theory is interpretable in another. Metatheoretical results often require a meta-language or a stronger framework than the object theory they analyze, since they discuss proofs, models, and other entities outside the theory itself.

Prominent metatheorems include Gödel's incompleteness theorems, which show fundamental limits of sufficiently strong formal systems; Gödel's completeness theorem, a metatheorem about first-order logic that connects semantic truth and syntactic provability; and relative-consistency or conservativity results, such as proving that the consistency of arithmetic follows from the consistency of a stronger theory. These results illuminate what formal theories can and cannot capture and guide foundational understanding.

Metatheorems are central to proof theory, model theory, and the philosophy of mathematics, aiding in the comparison of theories, the justification of formal methods, and the analysis of the foundations of mathematics and computer science.