Usually, we denote sets by capital letters $X,Y,\ldots$, while writing small letters $x,y,\ldots$ for their elements. In the Zermelo-Fraenkel set theory, sets can be themselves elements of other sets. Thus, there is no reason to introduce a different notation for sets and their elements and many sources in literature do without a distinctive notation. Nevertheless, we will keep sticking to this notation, just to make it clearer which sets are meant to be the elements and which sets are meant to contain these elements. When you read small letters $x,y,\ldots$ below, please keep in mind that they denote sets again.

Axiom: Zermelo-Fraenkel Axioms

A set is a mathematical object fulfilling the following axioms:

We will now explain the contents of each axiom separately and discuss how they improve the consistency of the set theory in comparison to the naive set definition.

  1. Axiom: Axiom of Existence
  2. Axiom: Axiom of Empty Set
  3. Axiom: Axiom of Extensionality
  4. Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)
  5. Axiom: Axiom of Pairing
  6. Axiom: Axiom of Union
  7. Axiom: Axiom of Power Set
  8. Axiom: Axiom of Foundation
  9. Axiom: Axiom of Infinity
  10. Axiom: Axiom of Replacement (Schema)
  11. Axiom: Axiom of Choice

Definitions: 1 2
Explanations: 3
Motivations: 4
Parts: 5 6
Proofs: 7

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  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
  2. Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition