Aczel's anti-foundation axiom


In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by, as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to a unique set. In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a Quine atom. A set theory obeying this axiom is necessarily a non-well-founded set theory.

Accessible pointed graphs

An accessible pointed graph is a directed graph with a distinguished vertex such that for any node in the graph there is at least one path in the directed graph from the root to that node.
The anti-foundation axiom postulates that each such directed graph corresponds to the membership structure of a unique set. For example, the directed graph with only one node and an edge from that node to itself corresponds to a set of the form x =.