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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]