Russel's Paradox

See: paradox

Question

If a set can contain sets as elements, can it contain itself? What is a set of elements that contains itself?

One book I read suggested the set of ideas is a set that contains itself.
- Prof. Nancy Day

Paradox

Consider the set whose elements are sets that do not contain themselves. We can write this in set builder notation:

is ?

proof

we just managed to show , which is only possible if the premises are inconsistent. Here, we have no premises. This is a problem!

Responses to Russel's Paradox

Arrange sets in a hierarchy in which sets at a level can only contain sets at a lower level

  1. Level 1: individual elements (scalars)
  2. Level 2: sets of individual elements
  3. Level 3: sets of sets of individual elements

Etc

Most common form: Zermelo-Fraenkel set theory

Thus the set cannot exist.