Intensional Set Theory

Publisher Summary There are two distinct notions of an effective-existence proof used in informal classical mathematics. This chapter introduces two kinds of variables, small ones ranging over explicitly- listed-hereditary finite sets and large ones ranging over all sets. The chapter describes the formal details of the system intensional set theory (IST), which includes formation rules, axioms, rules, the truth definition, the soundness theorem, existence and disjunction properties, and the extensions. The chapter also explores the challenge of extending IST by various true formulas so as to retain the possibility of a soundness theorem and its corollaries. Such formulas are essentially divided into two categories; in the first class, the formulas are true and become theorems of Zermelo-Fraenkel set theory (ZF) when all B's are deleted. There is no difficulty in carrying over the proof of the soundness theorem and its corollaries to extensions of IST by classically-valid formulas. In the second class of true formulas, those formulas are included which become false or at least unprovable in ZF if all B's are deleted.