1912, 1913), a monumental work of three volumes that
attempted to derive the whole of mathematics from
purely logical assumptions. Although not fully successful, the work was highly influential. Russell also discovered a fundamental paradox at the heart of basic SET
THEORY, today called RUSSELL’S PARADOX.
Russell studied mathematics and ethics at Trinity
College, Cambridge, and developed an interest in the
logical foundations of mathematics early in his career.
In 1903 he published Principles of Mathematics, a text
exploring the premise that all of mathematics could be
reduced to statements of logic, and that all mathematical proofs can be recast as proofs in the theory of logic.
It was during the writing of this work that Russell discovered his famous paradox in set theory, a discovery
that made it clear that more work was needed to find a
logical foundation to all of mathematics.
His next work, the famous Principia Mathematica,
was written as an attempt to extend the methods of his
Principles to a more general ramified theory that could
cope with all the disturbing self-referential paradoxes
that arose in set theory. Although recognized as a brilliant advancement in understanding the logical underpinnings of all of mathematics, the work still received
some criticism from the general mathematics community
as being either too ad hoc, too stringent, or too weak,
and that, at the very least, even more work was needed
to achieve the lofty goal Russell had set. In 1930 Austrian mathematician KURT GÖDEL (1906–78) shocked
the mathematical world by proving, in his incompleteness theorems, that there can be no logical base to all of
mathematics of the type Russell sought and that any
attempt to find a basic set of axioms on which to base
all of mathematics is a priori doomed to failure.
Russell’s influence in the development of mathematical logic was profound. He was elected to the
ROYAL SOCIETY in 1908 and wrote a number of stunning articles in the field throughout his career. Russell
was a political activist and was arrested several times
throughout his career for his antiwar activities. (His
1919 article “Introduction to Mathematical Philosophy” was written during a 6-month stint in prison.)
Russell taught at City College, New York, during the
1930s and was also a prominent figure in antiwar and
antinuclear protests in the United States.
Russell also wrote on a broad range of humanitarian topics, including political science, moral science,
and religious studies. He was awarded the Order of
Merit in 1949 and won the Nobel Prize for literature in
1950 for his collective writings championing “humanitarian ideals and freedom of thought.”
Russell died in Plas Penrhyn, Wales, on February 2,
1970.
Russell’s paradox (Russell’s antinomy) In 1902 mathematician and philosopher BERTRAND ARTHUR WILLIAM
RUSSELL (1872–1970) found a fundamental flaw at the
heart of basic SET THEORY. Usually sets are not members
of themselves. For instance, if D represents the set of
all dogs, then D, being a set, is not itself a dog. Let us call
such sets “normal.” Russell noted, however, that not
all sets are normal. For example, if S represents the set
of all sets, then S is itself a set and so belongs to S. The
set of all things that are not dogs is also nonnormal.
Let N represent the set of all normal sets. That is,
N is the set of all sets that are not members of themselves. Now ask: Is N normal? If the answer is yes, then
N is normal, making it an element of N, which is precisely what it means to be nonnormal. If the answer is
no, then N is nonnormal, meaning that N is an element
of N, the set of all normal sets. We have:
N normal ⇒ N nonnormal
N nonnormal ⇒ N normal
The set N can neither be normal nor nonnormal.
This contradiction is called Russell’s paradox, and
its discovery foiled mathematicians’ attempts to use set
theory as the foundational basis of all of mathematics.
It also suggested that all of mathematics might be
flawed if even the simplest of mathematical theories—
set theory—is fundamentally self-contradictory.
Russell also presented several alternative versions
of his paradox, including the famous “barber paradox”
that he presented in 1919:
A barber in a certain town has a sign that
reads, “I shave all those men, and only those
men, who do not shave themselves.” Who
shaves the barber?
In trying to answer this question, one is mired in the
same type of logical impasse as the original paradox.
In 1910, Russell, with the help of ALFRED NORTH
WHITEHEAD (1861–1947), published the first volume
of a mammoth three-volume treatise that attempted to
Russell’s paradox 453
attempted to derive the whole of mathematics from
purely logical assumptions. Although not fully successful, the work was highly influential. Russell also discovered a fundamental paradox at the heart of basic SET
THEORY, today called RUSSELL’S PARADOX.
Russell studied mathematics and ethics at Trinity
College, Cambridge, and developed an interest in the
logical foundations of mathematics early in his career.
In 1903 he published Principles of Mathematics, a text
exploring the premise that all of mathematics could be
reduced to statements of logic, and that all mathematical proofs can be recast as proofs in the theory of logic.
It was during the writing of this work that Russell discovered his famous paradox in set theory, a discovery
that made it clear that more work was needed to find a
logical foundation to all of mathematics.
His next work, the famous Principia Mathematica,
was written as an attempt to extend the methods of his
Principles to a more general ramified theory that could
cope with all the disturbing self-referential paradoxes
that arose in set theory. Although recognized as a brilliant advancement in understanding the logical underpinnings of all of mathematics, the work still received
some criticism from the general mathematics community
as being either too ad hoc, too stringent, or too weak,
and that, at the very least, even more work was needed
to achieve the lofty goal Russell had set. In 1930 Austrian mathematician KURT GÖDEL (1906–78) shocked
the mathematical world by proving, in his incompleteness theorems, that there can be no logical base to all of
mathematics of the type Russell sought and that any
attempt to find a basic set of axioms on which to base
all of mathematics is a priori doomed to failure.
Russell’s influence in the development of mathematical logic was profound. He was elected to the
ROYAL SOCIETY in 1908 and wrote a number of stunning articles in the field throughout his career. Russell
was a political activist and was arrested several times
throughout his career for his antiwar activities. (His
1919 article “Introduction to Mathematical Philosophy” was written during a 6-month stint in prison.)
Russell taught at City College, New York, during the
1930s and was also a prominent figure in antiwar and
antinuclear protests in the United States.
Russell also wrote on a broad range of humanitarian topics, including political science, moral science,
and religious studies. He was awarded the Order of
Merit in 1949 and won the Nobel Prize for literature in
1950 for his collective writings championing “humanitarian ideals and freedom of thought.”
Russell died in Plas Penrhyn, Wales, on February 2,
1970.
Russell’s paradox (Russell’s antinomy) In 1902 mathematician and philosopher BERTRAND ARTHUR WILLIAM
RUSSELL (1872–1970) found a fundamental flaw at the
heart of basic SET THEORY. Usually sets are not members
of themselves. For instance, if D represents the set of
all dogs, then D, being a set, is not itself a dog. Let us call
such sets “normal.” Russell noted, however, that not
all sets are normal. For example, if S represents the set
of all sets, then S is itself a set and so belongs to S. The
set of all things that are not dogs is also nonnormal.
Let N represent the set of all normal sets. That is,
N is the set of all sets that are not members of themselves. Now ask: Is N normal? If the answer is yes, then
N is normal, making it an element of N, which is precisely what it means to be nonnormal. If the answer is
no, then N is nonnormal, meaning that N is an element
of N, the set of all normal sets. We have:
N normal ⇒ N nonnormal
N nonnormal ⇒ N normal
The set N can neither be normal nor nonnormal.
This contradiction is called Russell’s paradox, and
its discovery foiled mathematicians’ attempts to use set
theory as the foundational basis of all of mathematics.
It also suggested that all of mathematics might be
flawed if even the simplest of mathematical theories—
set theory—is fundamentally self-contradictory.
Russell also presented several alternative versions
of his paradox, including the famous “barber paradox”
that he presented in 1919:
A barber in a certain town has a sign that
reads, “I shave all those men, and only those
men, who do not shave themselves.” Who
shaves the barber?
In trying to answer this question, one is mired in the
same type of logical impasse as the original paradox.
In 1910, Russell, with the help of ALFRED NORTH
WHITEHEAD (1861–1947), published the first volume
of a mammoth three-volume treatise that attempted to
Russell’s paradox 453