Interpretation of Constructive Multi-Typed Theory in the Theory of Arithmetical Truth

This paper considers an axiomatic theory BT, which can be used to formalise constructive mathematics. BT has an intuitionistic logic, combinatorial operations and sets of many types. BT has such constructive features as a predicative comprehension axiom and consistency with the formal Church thesis but BT is also consistent with classical logic. In addition to the properties of BT studied before, in this paper we study the proof-theoretical strength of BT and its fragments by constructing an interpretation of BT in a so called theory of arithmetical truth PATr, which is obtained from the Peano arithmetic PA by adding infinitely many truth predicates.