amorfset

Amorfset, often called an amorphous set, is a concept in set theory describing a special kind of infinite set with strong partition properties. In models of ZF set theory that do not assume the axiom of choice, amorfsets can exist; in contrast, under the axiom of choice (AC) every infinite set can be partitioned into two infinite subsets, so amorfsets do not exist there. Thus the existence of amorfsets is tied to the failure of AC.

Definition and basic properties: An amorfset A is an infinite set such that for any partition of A into two disjoint subsets A = B ∪ C, at least one of B or C is finite. Equivalently, every subset of A is either finite or cofinite in A (its complement in A is finite). Amorfsets are Dedekind-finite: there is no bijection between A and any proper subset of A. They are not well-orderable in standard frameworks where AC fails, reflecting the breakdown of typical well-ordering results in models containing amorfsets.

Existence and construction: Amorfsets are known to exist in certain models of ZF without AC, such as permutation models built from atoms (sometimes referred to in Fraenkel–Mostowski style constructions). Their existence demonstrates that some familiar consequences of AC, like the ability to partition every infinite set into two infinite parts, can fail.

Variants and terminology: There are refinements such as strongly amorphous sets, where every partition of the set into finitely many pieces yields exactly one infinite part and the remaining parts are finite. Weaker or related notions discuss how partitions behave under different finiteness constraints.

See also: Axiom of choice, ZF set theory, Dedekind-finite sets, permutation models, Fraenkel–Mostowski construction.