Non-well-founded Set Theory
Non-well-founded set theories are variants of axiomatic set theory which allow sets to contain 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 Peter Aczel's hyperset theory in 1988.
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.
Read more about Non-well-founded Set Theory: Details, Applications
Famous quotes containing the words set and/or theory:
“What shall we say who have knowledge
Carried to the heart? Shall we take the act
To the grave? Shall we, more hopeful, set up the grave
In the house? The ravenous grave?”
—Allen Tate (1899–1979)
“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
—Thomas Nagel (b. 1938)