Russel's Paradox

Let’s define a false axiom.
Axiom Schema of Comprehension. If is a property, then there exists a set .

Consider the set whose elements are all those (and only those) sets that are not members of themselves: .

• Does belong to ?
  If belongs to , then is not a member of itself, and so . On the other hand, if , then belongs to . In either case, we have contradiction.

Thus we must conclude that is not a set, and we must revise the intuitive notion of a set.

To solve this parados we abandon the Schema of Comprehension and keep its weak version, the Schema of Separation.

Axiom Esquema de Separación. Si es una propiedad (con parámetro ), entonces para cualquier y existe un conjunto que contiene todo con la propiedad .

Enlace al original

This results provides: The set of all sets does not exists, its paradoxical.