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The notion of the set is of fundamental importance to the whole contemporary mathematics. In modern formal mathematical treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets.

The first development of set theory as a mathematical discipline was in the frame of the so called naive set theory^{1} at the end of the 19th century by Georg Cantor who was led to this concept in order to allow a consistent work with infinite sets. As the date of birth of the set theory is given December 7, 1873 when G.Cantor proved that the set of real number is uncountable. Cantor’s work should be considered as a completion of a long historical process, cf. [1] p.298 or .

Actually probably the first one who began to work with sets was B. Bolzano who in 1847 considered sets with the following definition:* ...an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference*. Bolzano coined the usage of the German word Menge that Cantor later used for objects of his theory.

Naïve set theory is intuitive and simple, but unfortunately leads very soon to controversial statements. In contradiction with the axiomatic set theory it relies on an informal understanding of **sets **as collections of objects, called the **elements** or **members** of the set, that is on a predicate indicating that a collection is a set and a relation type symbol to represent set membership.

Gottlob Frege [2] was the first to attempt an axiomatization of set theory. His system requires one axiom and one axiom scheme^{2}:

**Axiom of Extensionality**:

**Axiom of Comprehension**:

Here denotes a formula expressible in first-order logic that contains as a free variable.

Based on this minimal extension of the first-order logic the rest of the naïve set theory can be defined. The Axiom of Comprehension allows the familiar constructions:

**List notation for sets**: denotes a set such that

**Brace notation for sets**: denotes a set such that .

Note that the Axiom of Comprehension guarantees that given a property expressible in the first-order logic there is a set that satisfies the first conjunct and the second conjuct states that there is no more than one set satisfying the property . In other words, the two mentioned axioms guarantee the existence of a unique set satisfying the conditions characterized by the list enumeration or property .

We can continue along the known lines:

**Intersection of sets**:

**Union of sets**:

**Subsets**:

**Proper subsets**:

Instead of we write .

As mentioned, the naïve set theory is not consistent^{3}. In summer 1901 Bertrand Russell discovered a paradox named after him which shows that the naive set theory is contradictory. Its essence is as follows:

The Axiom of Comprehension allows to consider any predicate formulable within the first-order logic and the set membership relation with the conclusion that there is a set consisting of all the items that satisfy that formula. For instance, the formula , i.e. “*the set of all sets that do not contain themselves as members”* gives

For this implies along standard logic reasoning (note that not employing the Axiom of Extensionality) that , a contradiction. Since we know that first-order theory itself is consistent, the source of the trouble appears to be the Axiom of Comprehension. Consequently, the Axiom of Extensionality could remains the same, but the Comprehension Axiom should be replaced usually by a system of axioms that are however not as intuitively obvious as expected.

Russell’s paradox had big impact on the later development of mathematics. Frege said that it undermined the whole mathematics. Therefore Russell (together with N.Whitehead) later tried to bring mathematics back onto a firm logical basis in *Principa Mathematica.* To avoid the paradoxes a theory of types was introduced according to which it is impossible to say that a class is or is not a member of itself, a way which was not generally accepted and so other ways were proposed. Note that K.Gödel showed the limitations of any axiomatic theory and therefore a complete axiomatization of mathematics could never be achieved.

^{1} | The term naive set theory (in contrast with axiomatic set theory) became an established term at the end of the first half of 20th century. It was then popularized by P. Halmos' book, Naive Set Theory (1960). |

^{2} | An axiom schema is a set - usually infinite - of well formed formulae, each of which is taken to be an axiom. |

^{3} | Consistency of an axiom system means that is impossible to deduce within the system that a statement and its negative are both true. |

[1] | Ebbinghaus, H., et al. (3rd improved printing 1992). Zahlen. Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest: Springer-Verlag. |

[2] | Frege, G. (1893/1903). Grundgesetze der Arithmetik I, II. Jena: H.Pohle. |

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