section_path: History/Naive set theory
section_name: Naive set theory
paragraph_text: Naive set theory
  
  The foremost property of a set is that it can have elements, also called members. Two sets are equal when they have the same elements. More precisely, sets A and B are equal if every element of A is an element of B, and every element of B is an element of A; this property is called the extensionality of sets. As a consequence, e.g. {2, 4, 6} and {4, 6, 4, 2} represent the same set. Unlike sets, multisets can be distinguished by the number of occurrences of an element; e.g. and represent different multisets, while and are equal. Tuples can even be distinguished by element order; e.g. (2, 4, 6) and (6, 4, 2) represent different tuples.
  
  The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
  
  Naïve set theory defines a set as any well-defined collection of distinct elements, but problems arise from the vagueness of the term well-defined.