Local Constructive Set Theory and Inductive Definitions

Local Constructive Set Theory (LCST) is intended to be a local version of constructive set theory (CST). Constructive Set Theory is an open-ended set theoretical setting for constructive mathematics that is not committed to any particular brand of constructive mathematics and, by avoiding any built-in choice principles, is also acceptable in topos mathematics, the mathematics that can be carried out in an arbitrary 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 formulated in a many-sorted language that has various forms of sort including, for each sort α a power-sort Pα , the sort of all sets of elements of sort α . For each sort α there is a binary infix relation ∈α that takes two arguments, the first of sort α and the second of sort Pα . For each formula φ and each variable x of sort α , there is a comprehension term {x : α | φ} of sort Pα for which the following scheme holds.