Russell's Paradox

Theorem
The axiom of abstraction leads to a contradiction.

Also presented as
This result can also be presented as:
 * There is no set $A$ that has every set as its element.

Its proof follows the same lines: by assuming that such an $A$ exists, and considering the set $\set {x \in A: \map R x}$ where $\map R x$ is the property $x \notin x$.

The same conclusion is reached.

Also known as
This result is also known as Russell's antinomy.

Also see

 * Barber Paradox