Nonwellfounded

Nonwellfounded refers to a concept in set theory and logic that describes sets or structures that do not have a well-foundedness property. A well-founded structure is one where there is no infinite descending chain of elements. In simpler terms, if you start at any element and keep going "down" or to a preceding element, you must eventually reach a point where there are no more preceding elements.

Sets that are nonwellfounded violate this principle. This means there can be infinite loops or self-referential structures. For example, a set that contains itself, or a set that contains a set that contains itself, and so on, is nonwellfounded. The classic example of a nonwellfounded set is often expressed as x such that x is in x, or x such that x is in y and y is in x.

The study of nonwellfounded sets emerged to address paradoxes and limitations found in traditional set theory, particularly those related to self-reference. While standard axiomatic set theories like ZFC (Zermelo–Fraenkel set theory with the Axiom of Choice) explicitly exclude nonwellfounded sets through the Axiom of Regularity, alternative set theories have been developed that allow for their existence. These theories, such as Aczel's anti-foundation axiom, provide a framework for reasoning about such structures.

Nonwellfoundedness has implications in various fields, including computer science, where it can be used to model cyclic data structures, program semantics, and concurrent processes. It also has connections to philosophy, particularly in discussions of paradoxes and the nature of mathematical objects.