my forty years on his shoulders - Department of Mathematics

30Gödel wrote four fundamental papers concerning formal systems basedon intuitionistic logic: (Gödel 1932a), (Gödel 1933a), (Gödel 1933b), (Gödel 1958).(Gödel 1972) is a revised version of (Gödel 1958).In (Gödel 32a), Gödel proves that the intuitionistic propositional calculuscannot be viewed as a classical system with finitely many truth values. He showsthis by constructing an infinite descending chain of logics intermediate instrength between classical propositional calculus and intuitionistic propositionalcalculus. For more on intermediate logics, see (Hosoi, Ono 1973) and (Minari1983).In (Gödel 1933a), Gödel introduces his negative interpretation in the formof an interpretation of PA = Peano arithmetic in HA = Heyting arithmetic. HereHA is the corresponding version of PA = Peano arithmetic based on intuitionisticlogic. It can be axiomatized by taking the usual axioms and rules of intuitionisticpredicate logic, together with the axioms of PA as usual given. Of course, onemust be careful to present ordinary induction in the usual way, and not use theleast number principle.It is natural to isolate his negative interpretation in these two ways:a. An interpretation of classical propositional calculus in intuitionisticpropositional calculus.b. An interpretation of classical predicate calculus in intuitionisticpredicate calculus.In modern terms, it is convenient to use ⊥,¬,∨,∧,→. The interpretation forpropositional calculus inductively interprets⊥ as ⊥.