In mathematical logic, Russell's paradox, also known as Russell's antinomy, is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions.
Most sets commonly encountered are not members of themselves. For example, consider the set of all squares in a plane. This set is not itself a square in the plane, thus it is not a member of itself. Let us call a set "normal" if it is not a member of itself, and "abnormal" if it is a member of itself. Clearly every set must be either normal or abnormal. The set of squares in the plane is normal. In contrast, the complementary set that contains everything which is not a square in the plane is itself not a square in the plane, and so it is one of its own members and is therefore abnormal.
Now we consider the set of all normal sets, R, and try to determine whether R is normal or abnormal. If R were normal, it would be contained in the set of all normal sets (itself), and therefore be abnormal; on the other hand if R were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that R is neither normal nor abnormal: Russell's paradox.
From the principle of explosion of classical logic, any proposition can be proved from a contradiction. Therefore, the presence of contradictions like Russell's paradox in an axiomatic set theory is disastrous; since if any formula can be proven true it destroys the conventional meaning of truth and falsity. Further, since set theory was seen as the basis for an axiomatic development of all other branches of mathematics, Russell's paradox threatened the foundations of mathematics as a whole. This motivated a great deal of research around the turn of the 20th century to develop a consistent or contradiction-free set theory.
In any event, Kurt Gödel in 1930–31 proved that while the logic of much of Principia Mathematica, the book by Russell and Alfred North Whitehead, now known as first-order logic, is complete, Peano arithmetic is necessarily incomplete if it is consistent. This is very widely—though not universally—regarded as having shown the logicist program to be impossible to complete.
QUESTION I: A set is a collection of objects. Do we need any other concept to understand Russell's Paradox?
QUESTION II: How Barber's Paradox is related to Russell's Paradox?