Abstract The aim of this paper is to formulate and study two weak axiom systems for the conceptual framework of constructive set theory (CST). Arithmetical CST is just strong enough to represent the class of von Neumann natural numbers and its arithmetic so as to interpret Heyting Arithmetic. Rudimentary CST is a very weak subsystem that is just strong enough to represent a constructive version of Jensenʼs rudimentary set theoretic functions and their theory. The paper is a contribution to the study of formal systems for CST that capture significant stages in the development of constructive mathematics in CST.