 
Summary: Local Constructive Set Theory and Inductive
Definitions
Peter Aczel
1 Introduction
Local Constructive Set Theory (LCST) is intended to be a local version of con
structive set theory (CST). Constructive Set Theory is an openended set theoretical
setting for constructive mathematics that is not committed to any particular brand
of constructive mathematics and, by avoiding any builtin choice principles, is also
acceptable in topos mathematics, the mathematics that can be carried out in an arbi
trary topos with a natural numbers object. We refer the reader to [2] for any details,
not explained in this paper, concerning CST and the specific CST axiom systems
CZF and CZF+
CZF+REA.
CST provides a global set theoretical setting in the sense that there is a single
universe of all the mathematical objects that are in the range of the variables. By
contrast a local set theory avoids the use of any global universe but instead is formu
lated in a manysorted language that has various forms of sort including, for each
sort a powersort P, the sort of all sets of elements of sort . For each sort
there is a binary infix relation that takes two arguments, the first of sort and
the second of sort P. For each formula and each variable x of sort , there is a
