Russell’s Paradox
Jami Durkee
Valerie Toothman
Jason Prindell
What is it?
Russell's paradox (also known as Russell's
antinomy) was discovered by Bertrand Russel
in 1901. It showed that the naïve set theory
created by Georg Cantor (which states any
definable collection is a set) leads to a
contradiction.
• Russell discovered the paradox in May or June 1901. By
his own admission in his 1919 Introduction to
Mathematical Philosophy, he "attempted to discover
some flaw in Cantor's proof that there is no greatest
cardinal".
• He did on the other hand discover that the axioms
Frege was using to formalize his logic were
inconsistent. Russell questioned his theory to the point
where Frege eventually felt forced to abandon many of
his views about logic and mathematics. There have
been many attempts in the last century to avoid it.
• The same paradox had been discovered a year
before by Ernst Zermelo but he did not publish
the idea.
• By 1908, Zermelo created the Zermelo set
theory that avoids the paradox.
Bertrand Arthur William Russell
May 18, 1872-Feb. 2, 1970
• He was a British philosopher, logician, essayist and social critic
best known for his work in mathematical logic and analytic
philosophy.
• He was noted for his spirited anti-war and anti-nuclear
protests.
• He remained a prominent public figure until his death
His family
• His paternal grandfather, John Russell, 1st Earl
Russell, was the third son of John Russell,
6thDuke of Bedford, and had twice been asked by
Queen Victoria to form a government, serving her
as Prime Minister in the 1840s and 1860s.
• Russell's mother, Katharine Louisa was the
daughter of Edward Stanley, 2nd Baron Stanley of
Alderley, and the sister of Rosalind Howard,
Countess of Carlisle. Kate and Rosalind's mother
was one of the founders of Girton College,
Cambridge.
Deaths
• In 1874 his mother died of diphtheria and sister
died
• 1876 his father died of bronchitis after a long
period of depression
• His grandparents then got custody of him and his
brother
• 1878 his grandfather died
• By order of his grandmother, instead of being
sent to school he was taught by governesses and
tutors, and thus acquired a perfect knowledge of
French and German.
Wives
• 1894 marries Alys Pearsall Smith
• 1921 divorce from Alys and marries Dora Black
• In 1927 they started a school for young
children, which they carried on until 1932.
• 1935 divorces Dora
• 1936 marries Patricia (Peter) Helen Spence
• 1952 divorce from Patricia (Peter) and marries
Edith Finch
School and Honors
• He entered Trinity College in 1890
• He was fined 110 pounds and dismissed from
Trinity College as a professor as a result of antiwar protests in 1916.
• He was elected Fellow of the Royal Society in
1908
• In 1931 he became the third Earl Russell upon the
death of his brother.
• He was awarded the Order of Merit in 1949 and
the Nobel Prize for Literature in 1950
Politics
• 1907, 1922, 1923 runs for parliament and is
defeated
• 1918 he was imprisoned for five months as a
result of anti-war protests.
• 1961 imprisoned for one week in connection
with anti-nuclear protests.
Ernst Friedrich Ferdinand Zermelo
July 27, 1871-May 21 1953
• He was a German mathematician whose work had
major implications for the foundations of
mathematics and philosophy.
• He is known for his role in developing ZermeloFraenkel axiomatic set theory and his proof of the
well-ordering theorem.
What is Russell’s Paradox?
Let R be the set of all sets that are not
members of themselves. If R qualifies as a
member of itself, it would contradict its own
definition as a set containing all sets that are
not members of themselves. On the other
hand, if such a set is not a member of itself, it
would qualify as a member of itself by the
same definition. This contradiction is Russell's
paradox.
letR  { x | x  x }, thenR  R  R  R
Definitions
• Set: Georg Cantor said “a set is a gathering together into
a whole of definite, distinct objects of our perception
and of our thought – which are called elements of the
set.”
– Example of a set is C = {4, 2, 1, 3}
• Contradiction: a statement or proposition that
contradicts or denies another or itself and is logically
incongruous.
• biconditional statement: A biconditional statement is
defined to be true whenever both parts have the same
truth value.

– The biconditional operator is denoted by a double-headed
arrow

Barber Example
Russell's paradox is based on examples like this:
Consider a group of barbers who shave only those men who
do not shave themselves. Suppose there is a barber in this
collection who does not shave himself; then by the definition
of the collection, he must shave himself. But no barber in the
collection can shave himself. (If so, he would be a man who
does shave men who shave themselves.)
Has it been solved?
• Creating a solution to Russell's paradox does
not involve 'fixing' the paradox. The paradox
will always lead to a contradiction.
• In order to 'solve' the paradox, set theory
must deal with the issue by voiding the
premise of the paradox.
Dealing with the paradox
• Several methods have been used to try to
deal with the paradox, two of which are:
– Russell's type theory
– Zermelo's set theory
Russell’s Type Theory
• attempts to tackle the paradox by creating 'types‘
• The lowest tier (Tier 1) of types would be primitive
objects, including numbers, shapes, ducks, etc.
• The next tier would be sets of Tier 1 objects (e.g.sets of numbers, sets of ducks)
• The third tier would be sets of sets of objects (sets
of sets of ducks).
• Defining types this way removes the possibility of
self-referencing sets R, where R = { x : x is not in x }.
Zermelo-Fraenkel set theory (ZFC)
• The accepted set theory
• modified version of Zermelo set theory to deal
with Russell’s Paradox
• created from several axioms
• The axiom that deals with Russell's paradox is
the axiom of separation