The Deep Set

In set theory, a "deep set" refers to a set that contains sets within sets to an arbitrarily deep level of nesting. The formal definition can vary depending on the specific context, but the core concept revolves around recursion and self-reference. A set is considered "deep" if its elements can be other sets, those sets can contain further sets, and this nesting can continue indefinitely, potentially without a defined limit on the depth.

This characteristic distinguishes deep sets from more traditional notions of sets where elements are often treated as atomic or where the nesting is relatively shallow. While typical set theories like Zermelo-Fraenkel set theory (ZFC) allow for sets to contain other sets, they do not explicitly emphasize or restrict the potential depth of this nesting.

The idea of deep sets often arises in contexts related to:

• Recursion and Fixed Points: Deep sets can be constructed using recursive definitions, where a set is defined in terms of itself. This relates to the concept of fixed points in set theory.

• Non-Well-Founded Set Theory: Standard set theories like ZFC rely on the axiom of foundation (also known as the axiom of regularity), which prohibits infinite descending membership chains (e.g., ... ∈ x₂ ∈ x₁ ∈ x). Deep sets, particularly those constructed with self-reference, are more naturally accommodated in non-well-founded set theories, such as Aczel's anti-foundation axiom (AFA). AFA allows for sets where x ∈ x or more complex cyclic memberships.

• Modeling Self-Referential Systems: Deep sets can be used as a tool to model systems with self-referential or hierarchical structures, in areas such as computer science (e.g., representing nested data structures) or philosophy (e.g., modeling concepts of consciousness or meaning).

The properties and behavior of deep sets are explored more rigorously within the framework of non-well-founded set theories, as these theories provide the necessary axiomatic basis to handle the potentially paradoxical nature of sets that contain themselves or have infinitely deep nesting. The specific characteristics of these sets will depend on the axioms used in the underlying set theory.