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Bertrand Russell (18721970 ) was born in Trelleck, Wales. His parents died when he was
three years old. He was educated privately and went to Trinity College, Cambridge,
where he was a brilliant student of mathematics and philosophy. In 1900,
Russell became acquainted with the work of the Italian mathematician Peano,
which inspired him to write The Principles of Mathematics (1903),
expanded in collaboration with Alfred No 515f59f rth Whitehead into three volumes of Principia
Mathematica (191013). The research, which Russell did during this period
together with Whitehead and which is preserved in many books and essays,
establishes him as one of the founding fathers of modern analytical philosophy.
Throughout his life Russell has also been an extremely outspoken and aggressive
moralist in the rationalist tradition of Locke and Hume. His many essays, often
in the form of short reflections or observations on moral or psychological
topics, are written in a terse, vivid, and provocative style. His greatest
literary achievement has been his History of Western Philosophy (1946).
Russell's external career has been chequered. The descendant of one of the
great families of the Whig aristocracy, he has always delighted in standing up
for his radical convictions with wilful stubbornness. In 1916, he was deprived
of his lectureship at Trinity College, Cambridge, after his pacifist activities
had brought him into conflict with the government, but in 1946 he was reelected
a Fellow. In 1918, he even went to prison for six months, where he wrote his Introduction
to Mathematical Philosophy (1919). In 1920, Russell travelled in Russia
and, subsequently, taught philosophy at Peking for a year. He went to the
United States in 1938 and taught there for several years at various
universities. Lord Russell has been a Fellow of the Royal Society since 1908;
he succeeded to the earldom in 1931 and, in 1949, received the Order of Merit.
In recent years Lord Russell has been active in political organizations such as
the Campaign for Nuclear Disarmament and
other groups with similar aims. The first two volumes of his autobiography,
covering the years from 1872 to 1944, appeared in 1967 and 1968, respectively.
Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a great ocean of anguish, reaching to the very verge of despair.
I have sought love, first, because it brings ecstasy  ecstasy so great that I would often have sacrificed all the rest of life for a few hours of this joy. I have sought it, next, because it relieves lonelinessthat terrible loneliness in which one shivering consciousness looks over the rim of the world into the cold unfathomable lifeless abyss. I have sought it finally, because in the union of love I have seen, in a mystic miniature, the prefiguring vision of the heaven that saints and poets have imagined. This is what I sought, and though it might seem too good for human life, this is whatat lastI have found.
With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved.
Love and knowledge, so far as they were possible, led upward toward the heavens. But always pity brought me back to earth. Echoes of cries of pain reverberate in my heart. Children in famine, victims tortured by oppressors, helpless old people a burden to their sons, and the whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer.
This has been my life. I have found it worth living, and would gladly live it again if the chance were offered me.
Russell's paradox is the most famous of the logical or settheoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.
Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all nonteacups, are members of themselves. Call the set of all sets that are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox prompted much work in logic, set theory and the philosophy and foundations of mathematics during the early part of the twentieth century.
Russell discovered his paradox in May of 1901 while working on his Principles of Mathematics (1903). Cesare BuraliForti, an assistant to Giuseppe Peano, had discovered a similar antinomy in 1897 when he noticed that since the set of ordinals is wellordered, it, too, must have an ordinal. However, this ordinal must be both an element of the set of ordinals and yet greater than any such element.
Russell wrote to Gottlob Frege with news of his paradox on June 16, 1902. The paradox was of significance to Frege's logical work since, in effect, it showed that the axioms Frege was using to formalize his logic were inconsistent. Specifically, Frege's Rule V, which states that two sets are equal if and only if their corresponding functions coincide in values for all possible arguments, requires that an expression such as f(x) may be considered to be both a function of the argument f and a function of the argument x. In effect, it was this ambiguity that allowed Russell to construct S in such a way that it could be a member of itself.
Russell's letter arrived just as the second volume of Frege's Grundgesetze der Arithmetik (The Basic Laws of Arithmetic, 1893, 1903) was in press. Immediately appreciating the difficulty that the paradox posed, Frege hastily added an appendix to the Grundgesetze which discussed Russell's discovery. Nevertheless, he eventually felt forced to abandon many of his views as a result of the paradox. Russell himself first discusses the paradox in detail in an appendix to his Principles of Mathematics.
The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. (For example, assuming both P and ~P, we can prove any arbitrary Q as follows: from P we can obtain P Q by the rule of Addition, and then from P Q and ~P we can obtain Q by the rule of Disjunctive Syllogism.) In the eyes of many, it therefore appeared that no mathematical proof could be trusted if it was discovered that the logic and set theory apparently underlying all of mathematics was contradictory.
Russell's paradox stems from the idea that any coherent condition may be used to determine a set. Attempts at resolving the paradox therefore have typically concentrated on various means of restricting the principles governing the existence of sets. Naive set theory contained the socalled unrestricted comprehension (or abstraction) axiom. This is an axiom, first introduced by Georg Cantor, to the effect that any predicate expression, P(x), containing x as a free variable will determine a set. The set's members will be exactly those objects that satisfy P(x), namely every x that is P. It is now generally agreed that such an axiom must be either abandoned or modified.
Russell's response to the paradox is contained in his theory of types. His basic idea is that we can avoid reference to S (the set of all sets that are not members of themselves) by arranging all sentences into a hierarchy. This hierarchy will consist of sentences (at the lowest level) about individuals, sentences (at the next lowest level) about sets of individuals, sentences (at the next lowest level) about sets of sets of individuals, etc. It is then possible to refer to all objects for which a given condition (or predicate) holds only if they are all at the same level or of the same "type". Although Russell first introduced the idea of types in his Principles of Mathematics, the theory found its mature expression five years later in his 1908 article "Mathematical Logic as Based on the Theory of Types" and in the monumental work he coauthored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913). In its details, Russell's type theory thus came to admit of two versions, the "simple theory" and the "ramified theory". Both versions have been criticized for being too ad hoc to eliminate the paradox successfully.
Other responses to the paradox include those of David Hilbert and the formalists (whose basic idea was to allow the use of only finite, welldefined and constructible objects, together with rules of inference that were deemed to be absolutely certain), and of Luitzen Brouwer and the intuitionists (whose basic idea was that one cannot assert the existence of a mathematical object unless one can also indicate how to go about constructing it).
Yet a fourth response to the paradox was Ernst Zermelo's 1908 axiomatization of set theory. Zermelo's axioms were designed to resolve Russell's paradox by restricting Cantor's naive comprehension principle. ZF, the axiomatization generally used today, is a modification of Zermelo's theory developed primarily by Abraham Fraenkel.
These four responses to the paradox have helped logicians develop an explicit awareness of the nature of formal systems and of the kinds of metalogical results that are today commonly associated with them.
He found it He
achieved only a little of it Pity
brought him back to heart Love
and knowledge led him upward toward the heavens Because of because To understand Pity for the suffering of mankind Longing for love Search for knowledge Was
governed by three passions:
THIS WAS IS LIFE AND HE FOUND IT WORTH
LIVING He
longed to alleviate the evil but he couldn't
RUSSELL'S
LIFE (18721970)

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