A constructive approach to nonstandard analysis

In the present paper we introduce a constructive theory of nonstandard arithmetic in higher types. The theory is intended as a framework for developing elementary nonstandard analysis constructively. More specifically, the theory introduced is a conservative extension of HA ” + AC. A predicate for distinguishing standard objects is added as in Nelson’s internal set theory. Weak transfer and idealisation principles are proved from the axioms. Finally, the use of the theory is illustrated by extending Bishop’s constructive analysis with infinitesimals. 1