**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"**.

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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