Dialectica interpretation [SEP] discoverer or inventor
Dialectica interpretation
In proof theory, the Dialectica interpretation is a proof interpretation of intuitionistic arithmetic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed by Kurt Gödel to provide a consistency proof of arithmetic. The name of the interpretation comes from the journal "Dialectica", where Gödel's paper was published in a 1958 special issue dedicated to Paul Bernays on his 70th birthday.
Motivation.
Via the Gödel–Gentzen negative translation, the consistency of classical
- Axiom of choice
- Markov's principle
- Independence of premise for universal formulas
is necessary and sufficient for characterising the formulas of HA which are interpretable by the Dialectica interpretation.
Extensions of basic interpretation.
The basic dialectica interpretation of intuitionistic logic has been extended to various stronger systems. Intuitively, the dialectica interpretation can be applied to a stronger system, as long as the dialectica interpretation of the extra principle can be witnessed by terms in the system T (or an extension of system T).