Aczel's anti-foundation axiom

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In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel (1988), 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[edit]

An accessible pointed graph is a directed graph with a distinguished vertex (the "root") 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 = {x}.

See also[edit]


  • Aczel, Peter (1988). Non-well-founded sets (PDF). CSLI Lecture Notes. 14. Stanford, CA: Stanford University, Center for the Study of Language and Information. ISBN 0-937073-22-9. MR 0940014. Archived from the original (PDF) on 2016-10-17. Retrieved 2008-03-12.
  • Goertzel, Ben (1994). "Self-Generating Systems". Chaotic Logic: Language, Thought and Reality From the Perspective of Complex Systems Science. Plenum Press. ISBN 978-0-306-44690-0. Retrieved 2007-01-15.
  • Akman, Varol; Pakkan, Mujdat (1996). "Nonstandard set theories and information management" (PDF). Journal of Intelligent Information Systems. 6 (1): 5–31. doi:10.1007/BF00712384.