Ekstensionality

Extensionality is a fundamental concept in logic and mathematics, particularly in set theory. It refers to the principle that a set is entirely determined by its elements. In other words, two sets are considered equal if and only if they contain exactly the same members. The order in which elements are listed or the way in which a set is defined does not affect its identity. For example, the set {1, 2, 3} is identical to the set {3, 1, 2} and also to the set {x | x is a positive integer less than 4}. The concept of extensionality is crucial for establishing a consistent and unambiguous foundation for set theory.

This principle is often contrasted with intentionality, which considers the defining properties or characteristics of a set rather than its actual contents. While extensionality focuses on "what is in the set," intentionality focuses on "how the set is described." In many contexts, especially when dealing with mathematical objects like sets, extensionality is the standard interpretation. It allows for precise reasoning and avoids ambiguities that could arise from different descriptions of the same collection of objects. The axiom of extensionality is a foundational axiom in Zermelo-Fraenkel set theory, underscoring its importance.