# 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]

## References[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.