Non-well-founded set theory

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 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 hyperset theory in 1988.^[1][2][3]

The theory of non-well-founded sets has been applied in the logical modelling of non-terminating 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.^[4]