On the independence of the well-ordering theorem from the ordering principle

Abstract

Publisher Summary This chapter focuses on the problem of the independence of the axiom of choice from the ordering principle. The chapter discusses basic notions for the axiom system: individuum, class, ɛ, A. Individuum and class are predicate names with one free variable; ɛ is the name of a relation which can hold between two individual or between an individuum and a class; A is an individual constant. As a logical basis for the subsequent system of axioms it is quite sufficient to take the narrower functional calculus. The chapter describes framework of von Neumann's system of set theory. It uses Latin letters to denote sets, and Gothic letters to denote domains (*) of sets. The letters ξ, η, ζ, τ denote ordinal numbers and the letters φ, ψ, א denote certain functions. Moreover, the usual notation of set theory is used. The chapter provides the proof of the main theorem.