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Local Constructive Set Theory and Inductive Definitions

Summary: Local Constructive Set Theory and Inductive
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 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 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+
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 many-sorted language that has various forms of sort including, for each
sort a power-sort 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


Source: Aczel, Peter - Departments of Mathematics & Computer Science, University of Manchester


Collections: Mathematics