How does Zermelo-Fraenkel set theory differ from Zermelo-Fraenkel set theory with the axiom of choice?

In the context of set theory, the primary difference between Zermelo-Fraenkel set theory (ZF) and Zermelo-Fraenkel set theory with the axiom of choice (ZFC) is the inclusion of the axiom of choice.12

The axiom of choice asserts that for any collection of nonempty sets, it is possible to form a new set by simultaneously selecting one element from each member of the collection, even if no specific rule or algorithm exists for the selection. While this is intuitive for finite sets, it is a powerful and nonconstructive tool when applied to infinite collections.32

Key distinctions include:

• Terminology: ZF refers to the system without the axiom of choice, while ZFC includes it.2
• Mathematical Consequences: The axiom of choice is equivalent to the well-ordering theorem and Zorn’s lemma. It allows for the definition of cardinal numbers as specific ordinals and simplifies the arithmetic of infinite cardinals.324
• Independence: It has been proven that the axiom of choice is independent of the other ZF axioms; it can neither be proved nor disproved within the ZF system, assuming ZF is consistent.3
• Controversy: Due to its nonconstructive nature and counterintuitive results—such as the Banach-Tarski paradox—some mathematicians have historically expressed unease regarding its use, though it is now considered an indispensable tool by the vast majority of the mathematical community.32