The Relation Reflection Scheme

In this paper we introduce a new axiom scheme, the Relation Reflection Scheme (RRS), for constructive set theory. Constructive set theory is an extensional set theoretical setting for constructive mathematics. A formal system for constructive set theory was first introduced by Myhill in [8]. In [1, 2, 3] I introduced a formal system CZF that is closely related to Myhill’s formal system and gave a natural interpretation of CZF and extensions of it in Martin-Löf’s constructive type theory, [7]. The axiom system CZF can be formulated in the same first order language as that of ZF, but uses intuitionistic logic rather than classical logic. But when the law of excluded middle is added the resulting classical theory has the same theorems as ZF. So, from the classical point of view, CZF does not involve any choice principle. The axiom CC of countable choice and even the stronger axiom DC of dependent choices have been accepted principles of constructive mathematics that have played an important role in the development of constructive analysis, particularly Bishop style constructive mathematics. In fact the interpretation of CZF in constructive type theory actually also gives an interpretation of each instance of the axiom scheme RDC of relative dependent choices; an axiom scheme that implies DC. So it is natural to consider allowing the free use of RDC in constructive set theory. But there are reasons for avoiding the use of any choice principles, even of CC, when possible. The Boolean-valued models of ZF generalise to Heyting-valued models of CZF. More precisely, in CZF, given a set-presented frame, which will form a complete Heyting algebra, Ω, there is a natural Ω-valued model construction that gives a reinterpretation of CZF, [5, 6], and, in general, these reinterpretations do not