Aczel's anti-foundation axiom

Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel in Aczel (1988). It states that every accessible pointed directed graph corresponds to a unique set. In particular, the graph consisting of a single vertex with a loop corresponds to a set which contains only itself as element, i.e. a Quine atom.

Accessible pointed graphs

An accessible pointed graph is a directed graph with a distinguished element (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 antifoundation 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}. A directed cycle graph of length 2 corresponds to a set of the form x = { {x} }.

See also

• Axiom of foundation
• von Neumann universe
• Non-well-founded set theory

References

• Aczel, Peter (1988). Non-well-founded sets.. CSLI Lecture Notes. 14. Stanford, CA: Stanford University, Center for the Study of Language and Information. pp. xx+137. ISBN 0-937073-22-9. MR0940014. http://standish.stanford.edu/pdf/00000056.pdf 
• 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. http://www.goertzel.org/books/logic/chapter_seven.htm. Retrieved 2007-01-15. 
• Varol Akman, Mujdat Pakkan (PDF). Nonstandard Set Theories and Information Management. http://www.cs.bilkent.edu.tr/~akman/jour-papers/jiis/jiis.pdf. Retrieved 2007-01-15.