Hypersets

Hypersets are sets that may contain themselves or participate in membership cycles, in contrast to the well-founded sets of standard set theory where no such cycles occur. They arise in non-well-founded set theory, a generalization of conventional set theory that allows self-reference and circular structures.

The most studied framework for hypersets is Aczel's Anti-Foundation Axiom (AFA). Under AFA, every directed graph where nodes represent potential sets and edges indicate membership corresponds to a unique set, and vice versa. This eliminates the need for the Foundation axiom to guarantee uniqueness and existence of sets, replacing it with a correspondence between graphs and sets. In this view, a hyperset can be described by its membership graph rather than by a finite construction.

Examples illustrate the difference from classical sets. A Quine atom is a hyperset Q with Q = {Q}, a self-membered set. More complex cycles, such as two sets A and B with A = {B} and B = {A}, are also hypersets. In a well-founded universe, such objects cannot exist, but in a non-well-founded framework they do and are well-behaved under AFA.

Hypersets are often represented and studied via graph-theoretic methods. Each directed graph corresponds to a unique hyperset, and the usual Mostowski collapse extends to non-well-founded graphs under AFA. Applications appear in computer science for modeling circular data structures and in formal semantics where self-reference and mutual containment are natural.

Historically, hypersets were developed to capture non-terminating or self-referential phenomena within a rigorous set-theoretic setting, with Aczel introducing the Anti-Foundation Axiom in the late 1980s.