Zermelo's Well-Ordering Theorem

Zermelo's Well-Ordering Theorem states that every set, no matter how complex or large, can be arranged in a complete, ordered sequence where each element has a unique position—called a well-ordering. This means you can "line up" all elements so that every subset has a smallest element, making the set comparable like natural numbers. The theorem relies on the Axiom of Choice, which allows such an ordering to be constructed even for highly abstract or infinite sets, enabling mathematicians to work with all sets in a consistent, ordered framework.