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Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
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Russell paradox

Formulation of the Russell's paradox

Illustrations of the Russell's paradox

Russell's paradox and naive set theory

Avoiding Russell's paradox with type theory

Avoiding Russell's paradox with axiomatic set theory

Formulation of the Russell's paradox

Russell's paradox: The set M is the set of all sets that do not contain themselves as members. Does M contain itself?

If it does, it is not a member of M according to the definition.
If it does not, then it has to be a member of M, again according to the definition of M.
Therefore, the statements "M is a member of M " and "M is not a member of M " both lead to contradictions.

There are some versions of Russell's paradox, for example: "A is an element of M if and only if A is not an element of A".

Russel paradox

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Illustrations of the Russell's paradox

1) List of all lists that do not contain themselves.
If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself.

2) Barber paradox
The paradox considers a town with a male barber who shaves all and only those men who do not shave themselves.
The question is: Who shaves the barber?

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Russell's paradox and naive set theory

The paradox was discovered by Bertrand Russell in 1901. The paradox arises within naive set theory. It showed that naive set theory (set theory as it was used by Georg Cantor and Gottlob Frege) contained contradictions. After the discovery of the paradox, it becomes clear that naive set theory must be replaced by something in which the paradoxes can't arise. Two solutions were proposed: type theory and axiomatic set theory.

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Avoiding Russell's paradox with type theory

Russell himself, together with Whitehead proposed a type theory, in which sentences were arranged hierarchically. This avoids the possibility of having to talk about the set of all sets that are not members of themselves, because the two parts of the sentence are of different types - that is, at different levels.

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Avoiding Russell's paradox with axiomatic set theory

Another approach to avoid such types of paradoxes was an axiomatic set theory, proposed by Ernst Zermelo. This theory determines what operations were allowed and when. The set of all sets M cannot be constructed like that and is not a set in this theory.

by Tetyana Butler

Complex functions Tutorial
Complex analysis is studying the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of algebra and trigonometry of real numbers may break down when applied to complex numbers.
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