About the Axiom of Choice

Publisher Summary The axiom of choice is crucial not only in logic (set theory and model theory) but also in other modern disciplines as well such as point set topology, algebra, functional analysis, and measure theory. This chapter presents examples of fundamental theorems of abstract algebra and topology whose proofs use the axiom of choice. In some instances, the theorems are as strong as the axiom of choice—an example of a statement equivalent to the axiom of choice is the Tychonoff product theorem in point set topology. Some objections to the axiom of choice are based on the fact that the axiom has paradoxical consequences. The most famous example is Banach–Tarspkai paradox. The Banach–Tarspkai paradox states that using the axiom of choice, one can cut a ball into a finite number of pieces that can be so rearranged that one obtains two balls of the same size as the original ball. The chapter also sketches the proof of this paradox to show how the axiom of choice is used and that there is nothing paradoxical about this theorem.