Interpretability Logic

Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbert-style programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove conservation results within a properly weak theory. An advantage of using interpretations is that even if their use should -perhaps- be classified as a prooftheoretical method, it is often possible to employ a model-theoretical heuristics. An example is given in section 7.2 where a conservation result due to Paris & Wilkie, which is proven by a model-theoretical argument, is formalized in a weak theory. For more discussion of and perspective on the use of interpretability in reductive programs the reader is referred to Feferman[1988]