  In Feferman's work, explicit mathematics and theories of generalized
inductive definitions play a central role. One objective of this article is to
describe the connections with Martin-Lof type theory and constructive
Zermelo-Fraenkel set theory. Proof theory has contributed to a deeper grasp of
the relationship between different frameworks for constructive mathematics.
Some of the reductions are known only through ordinal-theoretic
characterizations. The paper also addresses the strength of Voevodsky's
univalence axiom. A further goal is to investigate the strength of
intuitionistic theories of generalized inductive definitions in the framework
of intuitionistic explicit mathematics that lie beyond the reach of Martin-Lof
type theory.
