Publisher Summary This chapter reviews certain logical notions, such as propositional functions, introduces axioms, and forms the bases of the elucidation of set theory. Not all mathematicians accept the axiom of choice without reserve; some of them view this axiom with a certain measure of distrust. They claim that the proofs involving the axiom of choice are of a different nature from the proofs not involving it because it predicates the existence of a set B without giving a method of constructing it. Among the consequences of the axiom of choice, there are many very peculiar ones. Axioms in mathematical theories can play one of two roles. There are cases where axioms completely characterize the theory; that is, they constitute in some sense a definition of the primitive notions of the theory. In other cases, axioms formalize only certain chosen properties of the primitive notions of the theory. Some consequences of the axioms are highlighted in the chapter.