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Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
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Set theory paradoxes
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Set theory paradoxes

Naive set theory and paradoxes

Avoiding paradoxes with type theory

Avoiding paradoxes with axiomatic set theory

Naive set theory and paradoxes

Set theory was created at the end of the 19-th century by Georg Cantor in order to allow to work with infinite sets consistently, and attempted to formalize the set using a minimal collection of independent axioms. Later it was named "naive set theory".

In 1901 Bertrand Russell discovered paradox, which was named Russell's paradox. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox. After the discovery of the paradox, it becomes clear that assuming that one could perform any operations on sets without restriction led to paradoxes, and 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 paradoxes with type theory

To avoid Russell's paradox Russell himself, together with Whitehead proposed a theory of types in which sentences were arranged hierarchically. At the lowest level are sentences about individuals. At the next level are sentences about sets of individuals; at the next level, sentences about sets of sets of individuals, and so on. It is then possible to refer to all objects for which a given condition (or predicate) holds only if they are all at the same level or of the same "type". 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.
Theory of type have been criticized for being too ad hoc to eliminate the paradox successfully.

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Avoiding paradoxes with axiomatic set theory

To avoid Russell's paradox and other related problems Ernst Zermelo proposed an axiomatic set theory. Zermeloís axioms were designed to resolve Russellís paradox by restricting Cantorís naive comprehension principle. This theory determines what operations were allowed and when. Today Zermelo-Fraenkel set theory is the most common version of the axiomatic set theory. Theory does not assume that, for every property, there is a set of all things satisfying that property. It assumes that for any given set and any definable property, there is a subset of all elements of the given set satisfying the property. 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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