my forty years on his shoulders - Department of Mathematics

33the result of simultaneously replacing every atomic subformula ψ of ϕ by (ψ ∨A). In particular, ⊥ gets replaced by what amounts to A.The A translation is an interpretation of HA in HA. I.e., if ϕ A is defined,and HA proves A, then HA proves ϕ A . Also, obviously HA proves A → ϕ A .Now suppose (∃n)(F(n,m) = 0) is provable in PA, where F is a primitiverecursive function symbol. By Gödel’s negative interpretation, ¬¬(∃n)(F(n,m) =0) is provable in HA. Write this as ((∃n)(F(n,m) = 0) → ⊥) → ⊥.By taking the A translation, with A = (∃n)(F(n,m) = 0), we obtain that HAproves((∃n)(F(n,m) = 0 ∨ (∃n)(F(n,m) = 0)) → (∃n)(F(n,m) = 0)) → (∃n)(F(n,m) = 0.((∃n)(F(n,m) = 0) → (∃n)(F(n,m) = 0)) →(∃n)(F(n,m) = 0.(∃n)(F(n,m) = 0).This method applies to a large number of pairs T/T’ as indicated in(Friedman 1973) and (Leivant 1985).(Godel 1958) and (Godel 1972) present Gödel’s so called Dialecticainterpretation, or functional interpretation, of HA. Here HA = Heytingarithmetic, is the corresponding version of PA = Peano arithmetic withintuitionistic logic. It can be axiomatized by taking the usual axioms and rules ofintuitionistic predicate logic, together with the axioms of PA as usual given. Ofcourse, one must be careful to present ordinary induction in the usual way, andnot use the least number principle.In Gödel’s Dialectica interpretation, theorems of HA are interpreted asderivations in a quantifier free system T of primitive recursive functionals of