Proof Theory: Some Applications of Cut-Elimination

Gödel's Functional Interpretation and Its Use in Current Mathematics

“…So in order to perform the modus ponens under the no-counterexample interpretation one needs exactly the same use of bar recursion B 0,1 as discussed in 4.3 above (for a detailed discussion see [60]). For the special case (+), the no-counterexample interpretation yields results as strong as the combination of negative translation with functional interpretation but to verify the soundness of the former for a given system one either has to prove the soundness of the latter or to apply a suitable form of ε-substitution or cut-elimination ( [76,77,97]) which destroys the modularity of the interpretation.…”

“…In fact, the original proof of the no-counterexample interpretation (based on a different description of the type-2 functionals in T as the α(< ε 0 )-recursive functionals of type 2) as given in [76,77] was based on the ε-substitution method developed in [1] and so rather on a form of cut-elimination than established as a proof interpretation. A proof by direct use of cut-elimination was given subsequently in [97]. In fact, [34] shows that the no-counterexample interpretation was clearly anticipated by Gödel's analysis of Gentzen's consistency proof for PA based on cut-elimination.…”

Ordinal Analysis and the Infinite Ramsey Theorem

“…By employing a partial truth predicate for Π 0 n -formulae and transfinite induction up to ε k+1 , one shows that PA + TI(≺ ε ω ) A (cf. [10]). …”

Reflection principles and provability algebras in formal arithmetic

“…[78]). There also is asimple (but less constructive) alternative model-theoretic proof based on compactness theorem.…”

“…We rely on the standard cut-elimination techniques for a variant of sequent calculus due to Tait that is presented, for example, in Schwichtenberg [78]. Formulas in Tait's calculus are constructed as in the first order logic from atomic formulas and their negations by the connectives ∧, ∨ and the quantifiers ∀, ∃.…”