Naive set theory
Sets, membership and equality

In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite.

The definition of sets goes back to Georg Cantor. He wrote in his 1915 article Beiträge zur Begründung der transfiniten Mengenlehre:

“Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten unserer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen.” – Georg Cantor

“A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.” – Georg Cantor
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