# Friedman on interpretations

## Albert Visser

During the period 1975-1980, Harvey Friedman proved two surprising fundamental results concerning interpretations. Both results concern a special class of theories: the finitely axiomatized sequential theories.

The first result is known as *the Friedman Characterization.* Two salient ways of comparing theories
are *relative interpretability* and *verifiable relative consistency.* It is well known that generally these
two notions do not coincide. Friedman shows that for the finitely axiomatized sequential theories
the notions do coincide provided that we choose Elementary Arithmetic as the base theory to
verify relative consistency and that we consider cut-free consistency rather than ordinary consistency.

The second result concerns the relation between interpretability and faithful interpretability. We can easily provide examples to show that these notions do not coincide. However, Friedman shows that on the finitely axiomatized sequential theories the two notions do coincide.

The aim of this talk is primarily to explain the two results. We hope to give the hearer some feeling for the results and to provide some perspective on the results.