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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 $X=\emptyset.$

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

| | | | | created: 2014-03-22 15:09:49 | modified: 2018-12-09 13:09:12 | by: bookofproofs | references: [656], [7838]

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


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Bibliography (further reading)

[656] Hoffmann, Dirk W.: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

[7838] Kohar, Richard: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016

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