Editing Non-well-founded set theory (section)

{{Short description|Theory that allows sets to be elements of themselves}}
'''Non-well-founded set theories''' are variants of [[axiomatic set theory]] that allow sets to be elements of themselves and otherwise violate the rule of [[well-foundedness]]. In non-well-founded set theories, the [[axiom of regularity|foundation axiom]] of [[ZFC]] is replaced by axioms implying its negation.

The study of non-well-founded sets was initiated by [[Dmitry Mirimanoff]] in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an [[axiom]]. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until the book Non-Well-Founded Sets by [[Peter Aczel]] introduces [[Non-well-founded set theory#Applications|hyperset theory]] in 1988.{{sfnp|Pakkan|Akman|1994|loc=[http://tinf2.vub.ac.be/~dvermeir/mirrors/www.cs.bilkent.edu.tr/%257Eakman/jour-papers/air/node8.html section link]}}{{sfnp|Rathjen|2004|p=}}{{sfnp|Sangiorgi|2011|pp=17–19, 26}}

The theory of non-well-founded sets has been applied in the [[logic]]al [[model (abstract)|modelling]] of non-terminating [[Computing|computational]] processes in computer science ([[process algebra]] and [[final semantics]]), [[linguistics]] and [[natural language]] [[semantics]] ([[situation theory]]), philosophy (work on the [[Liar Paradox]]), and in a different setting, [[non-standard analysis]].{{sfnp|Ballard|Hrbáček|1992|p=}}