nonwellorderable

Nonwellorderable is a term used in set theory to describe a set that cannot be well-ordered. A set is well-orderable if there exists a relation that well-orders the elements, equivalently meaning there is a bijection between the set and some ordinal. When no such well-ordering exists, the set is called nonwellorderable.

The concept is tied to the Axiom of Choice (AC). AC states that every set can be well-ordered, so in any theory where AC holds, all sets are well-orderable. Conversely, the absence of AC allows the existence of nonwellorderable sets. Thus, proving the existence of a nonwellorderable set is a demonstration that AC fails in that mathematical universe; many statements about well-orderings become independent of ZF alone.

Examples and context: In certain models of ZF where the Axiom of Choice fails, there exist sets that cannot be put into a well-order. The real numbers, for instance, can be non-wellorderable in some models without AC. However, the exact status of specific sets can be model-dependent, and there is no single universal concrete example within ZF alone.

Related notions include the well-ordering theorem, which is equivalent to AC, and the study of ordinals and cardinals that classify well-orderable sets. Tools such as Hartogs numbers help demonstrate why some sets cannot be put into a bijection with any ordinal without invoking AC.

See also: Axiom of Choice, ZF, well-ordering theorem, Hartogs number.