Abstract
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 proof-theoretical 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].
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Visser, A. (1990). Interpretability Logic. In: Petkov, P.P. (eds) Mathematical Logic. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0609-2_13
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