Rudimentary and arithmetical constructive set theory

The CST conceptual framework is a set theoretical approach to constructive mathematics initiated by Myhill in [Myh75]. It has been given a philosophical foundation via formal interpretations into versions of Martin-Löf’s Intuitionistic Type Theory, [GA06, Acz86, Acz82, Acz78]. There are several axiom systems for Constructive Set Theory of varying logical strength. Perhaps the most familiar ones are CZF and CZF ≡ CZF + REA, see [AR01]. The axiom system CZF is formulated in the first order language L∈ for intuitionistic logic with equality having ∈, an infix binary relation symbol, as the only non-logical symbol. So the logical symbols are ⊥,∧,∨,→,∀,∃,=. We use the standard abbreviations for ↔ , ¬ and the bounded quantifiers (∀x ∈ t) and (∃x ∈ t). A formula is bounded if all its quantifiers are bounded. We assume a standard axiom system for intuitionistic logic with equality. The non-logical axioms and schemes of CZF are the axioms of Extensionality, Emptyset, Union, Pairing and Infinity and the axiom schemes of Bounded