ekstensionalitetsaxiomet

The axiom of extensionality is a fundamental principle in set theory. It states that two sets are equal if and only if they have the same elements. In simpler terms, the identity of a set is determined solely by its members, not by any other properties.

This axiom is crucial for defining sets precisely. For example, if we have a set A containing the elements {1, 2, 3} and a set B containing the elements {3, 1, 2}, the axiom of extensionality tells us that A and B are the same set because they contain exactly the same elements, regardless of the order in which they are listed.

Conversely, if set C contains the elements {1, 2, 3, 4}, then C is not equal to A or B, because it has an element (4) that A and B do not possess. The axiom of extensionality provides a clear and unambiguous criterion for set equality, which is essential for building a consistent foundation for mathematics. Without it, it would be impossible to determine whether two sets are indeed the same, leading to potential paradoxes and inconsistencies in set theory. It is a cornerstone of Zermelo-Fraenkel set theory (ZF) and its extension ZFC.