Buss and Cook's work on bounded arithmetic reveals that the metatheoretic reasoning validating transitivity of reductions may itself require proof-theoretic resources exceeding those formalizable in weak systems.
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Transitivity of reductions(the principle whose validity is in question)
If you can simplify problem A into problem B, and problem B into problem C, then you should be able to simplify A directly into C—a basic logical chain.
bounded arithmetic(Studied for the relationship between arithmetic theories and computational complexity.)
A family of first-order arithmetical theories whose levels correspond to the levels of the Polynomial Hierarchy, similar in form to systems such as Primitive Recursive Arithmetic and Peano Arithmetic but with quantifiers bounded by terms.