Chapter 6 - Constructivism and Choice

Abstract

The terms “construct” and “construction” are used by most mathematicians and these terms may be applied to any argument that builds something complicated from seemingly simpler things. An existence proof is constructive if the proof actually finds the object in question by a procedure involving many steps or, in some cases, if the proof approximates the object arbitrarily by a procedure involving many steps. Constructivists are mathematicians who study such proofs and/or who prefer such proofs; constructivism is the study of such proofs. This chapter introduces the Axiom of Choice (AC) and a few weakened forms of Choice. Conventional set theory is ZF + AC—that is, the Zermelo–Fraenkel set theory plus the Axiom of Choice. ZF is a formalization of intuition about sets. The chapter also defines several relations among these principles which are summarized in a chart. All assertions in the chart are understood to be in conjunction with ZF. Implications in the chart are downward and most of these implications can be proved by this chart. One might almost think of these principles as the “constructive component” and the “nonconstructive component” of AC.