In the historical development of set theory, it was mentioned that Russel demonstrated, the following classical definition is not sufficient since it leads to paradoxical constructs. If you are a beginning student of the set theory, the classical definition of Cantor is a good starting point, because it is highly intuitive.

Definition: Set, Set Element, Empty Set (Cantor) 

(Original, naive set definition of Cantor (1895))¹

A set is a combination of well-distinguishable, mathematical objects. Let \(X\) be a set.

• If an object \(x\) belongs to the set \(X\), it is called ist element and written as \(x\in X\).
• We write \(x\notin X\), if \(x\) is not an element of the set \(X\).
• If $X$ has no elements, we call $X$ empty, and write $X=\emptyset,$ otherwise non-empty and write $X\neq\emptyset.$

¹ Nowadays, we use the Zermelo-Fraenkel axioms (ZFA) to define sets.

| | | | | created: 2014-03-22 15:09:49 | modified: 2019-09-07 08:12:09 | by: bookofproofs | references: [656], [7838]

1.Explanation: Possibilities to Describe Sets, Venn-Diagrams, List, and Set-Builder Notations