**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