A hyperset, or a non-well-founded set, is in simplest terms a set which can contain itself as a member. In standard Zermelo-Fraenkel set theory (ZFC), the possibility of such sets existing is ruled out by the Axiom of Foundation. This axiom states that every nonempty set A has a member, m, such that the union of m and A is the empty set. Another interpretation of the Foundation Axiom is that there exists no infinite descending element-of sequences.
For roughly half a century, this axiom has been widely accepted as the only possible way to accurately describe the domain of sets. However, in recent years, some mathematicians have replaced the Foundation Axiom with an Antifoundation Axiom, without altering any of the other axioms of ZFC. By allowing for the possibility of set-in-itself membership, a new, entirely different domain of set theory has been created. One mathematician noted that the extension of the domain of sets in this manner (to include hypersets), is comparable to the extension of the rational number system to the reals. It is this extension which this paper attempts to analyze.