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...

binary relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty subset S ⊆ X has a minimal...

set theory Non-well-founded set theory List of first-order theories § Set theories Holmes, M. Randall. "Alternative Axiomatic Set Theories". Stanford...

exist in systems of set theory that include the axiom of regularity, but they can exist in non-well-founded set theory. ZF set theory with the axiom of...

way. A set theory containing a universal set is necessarily a non-well-founded set theory. The most widely studied set theory with a universal set is Willard...

(and well-foundedness) as an axiom to be observed by all sets; in later papers Mirimanoff also explored what are now called non-well-founded sets ("extraordinaire"...

mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset of...

in non-well-founded set theory. In this context, each vertex of an accessible pointed graph models a (non-well-founded) set within Aczel's (non-well-founded)...

by V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide...

framed in a non-well-founded set theory. One could think of the relation of situation theory to situation semantics as like that of type theory to Montague...