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Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
Complex functions paradoxes
Infinity paradoxes
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Barber paradox

Formulation of the Barber paradox

Barber paradox and naive set theory

Avoiding Barber paradox with type theory

Avoiding Barber paradox with axiomatic set theory

Formulation of the 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?

When one thinks about whether the barber should shave himself or not, the paradox begins to emerge.

If the barber does not shave himself, according to the rule he must shave himself.
If he does shave himself, according to the rule he will not shave himself.

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Barber paradox and naive set theory

This paradox is often attributed to Bertrand Russell. The paradox arises within naive set theory. It is used as illustration to famous Russell paradox, which Bertrand Russell devised to show that naive set theory (set theory as it was used by Georg Cantor and Gottlob Frege) contained contradictions. As it turned out, assuming that one could perform any operations on sets without restriction led to paradoxes. 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 Barber paradox with type theory

Russell himself, together with Whitehead proposed type theory in which sentences were arranged hierarchically. This resolves such types of paradoxes, because
1) a barber as a citizen of the town, who shaves himself
and
2) a barber as a professional, who shaves others
are of different types.

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Avoiding Barber 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. Such barber is not allowed 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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