This extension is needed for, among other things, modeling the concepts of " is an element of " or " is a subset of " in set theory, without running into logical inconsistencies such as Russell's paradox.

As it turned out, assuming that one can perform any operation on sets without restriction leads to paradoxes such as Russell's paradox and Berry's paradox.

Paradoxical versions of the Epimenides problem are closely related to a class of more difficult logical problems, including the liar paradox, Russell's paradox, and the Burali-Forti paradox, all of which have self-reference in common with Epimenides.

In that article, Russell uses the Epimenides paradox as the point of departure for discussions of other problems, including the Burali-Forti paradox and the paradox now called Russell's paradox.

He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory ( Burali-Forti paradox, Cantor's paradox, and Russell's paradox ) to a meeting of the Deutsche Mathematiker – Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

Dialetheism arises from formal logical paradoxes, such as the Liar's paradox and Russell's paradox.

For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.

These were avoided in PM by building an elaborate system of types: a set of elements is of a different type than is each of its elements ( a set is not an element ; one element is not the set ) and one cannot speak of the " set of all sets " and similar constructs, which would lead to paradoxes ( see Russell's paradox ).

Bertrand Russell is credited with noticing the existence of such paradoxes even in the best symbolic formalizations of mathematics in his day, in particular the paradox that came to be named after him, Russell's paradox.

Bertrand Russell invented the first type theory in response to his discovery that Gottlob Frege's version of naive set theory was afflicted with Russell's paradox.

It avoids Russell's paradox by first creating a hierarchy of types, then assigning each mathematical ( and possibly other ) entity to a type.

Origin of Russell's Theory of Types: In a letter to Gottlob Frege ( 1902 ) Russell announced his discovery of the paradox in Frege's Begriffsschrift.

* Russell's paradox

The International Bible Students Association ( precursor to Jehovah's Witnesses ) identified Abaddon as Satan in the 1917 seventh and final volume of Millennial Dawn, to which Charles Taze Russell's name was attached ( although his authorship of the work is doubted ).

However, Russell's theory analyses proper names into a logical structure which makes sense of this problem.

Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic.

Bill Clinton gave an emotional eulogy in which he recalled their boyhood times together and quoted a line from Leon Russell's " A Song for You ": " I love you in a place that has no space and time.

He often composed specifically for the style and skills of these individuals, such as " Jeep's Blues " for Johnny Hodges, " Concerto for Cootie " for Cootie Williams, which later became " Do Nothing Till You Hear from Me " with Bob Russell's lyrics, and " The Mooche " for Tricky Sam Nanton and Bubber Miley.

One of these biopics is Ken Russell's 1977 film, Valentino, in which Valentino is portrayed by Rudolf Nureyev.

It was suggested to him as an alternate form of Russell's paradox, which he had devised to show that set theory as it was used by Georg Cantor and Gottlob Frege contained contradictions.

* P. Craig Russell's Jungle Book Stories ( 1997 ) collects three stories, actually adapted from The Second Jungle Book, which originally appeared between 1985 and 1996.

Fame came with Jackson's starring role in the controversial Women in Love ( 1969 ) for which she won her first Academy Award for Best Actress, and another controversial role as Tchaikovsky's nymphomaniac wife in Ken Russell's The Music Lovers added to her image of being prepared to do almost anything for her art.

In 1989, she appeared in Ken Russell's The Rainbow, playing Anna Brangwen, mother of Gudrun, the part which had won her her first Academy Award.

Films in which Perkins appeared in which his character had much resonance with Norman Bates include Ken Russell's Crimes of Passion ( 1984 ) with Kathleen Turner, and the Hungarian-produced Jekyll-Hyde remake Edge of Sanity and Daughter of Darkness.

Lowe, Tulin, Williams and Weakley were introduced to David Hassinger, then resident engineer at RCA studios, who arranged for them to record some demos at Leon Russell's home recording facility ( which he called Sky Hill Studios ).

Thus the majority vote in favour of the amendment to the militia bill caused the downfall of Russell's ministry, which occurred on February 21, 1852.

Russell's fourth novel, Dreamers of the Day ( released in March 2008 ), revolves around the 1921 Cairo Peace Conference, which laid the foundations for the modern Middle East.

In the 1820s the pits became incorporated as Russell's Colliery, which then became The Wallsend and Hebburn Coal Company Ltd. By 1924 the colliery employed 2183 people.

East, at one point, had several couples, all of which were descended from one of Russell's dogs.

Ken Russell's 1971 film version of the show, starring Twiggy and Christopher Gable, was an alternative interpretation, weaving the basic plot into a more complicated story in which a seaside dramatic company, performing the show, is visited by a film producer ( Vladek Sheybal ) on the very night that the leading lady ( Glenda Jackson ) has to be replaced by her shy understudy Polly Browne ( Twiggy ).

Spencer split from other whiggish aristocratic Liberals in 1866 on the issue of Russell's reform bill, which he supported, and his loyalty was rewarded by his appointment as Lord Lieutenant of Ireland when Gladstone returned to power in 1868.

He first came to wider public attention when, along with Pauline Boty, Derek Boshier and Peter Phillips, he was featured in Ken Russell's Monitor film on pop art, Pop Goes the Easel, which was broadcast on BBC television in 1962.

The following quote from Bertrand Russell's " Theory of Knowledge " illustrates the difficulty in defining knowledge: " The question how knowledge should be defined is perhaps the most important and difficult of the three with which we shall deal.

This variation of Russell's paradox shows that no set contains everything.

* Ken Russell's The Devils shows the priest Urbain Grandier being tortured with a form of the boot.

In addition to his success as a character actor in classical drama, he was known for his work as a singer and comic actor in revue and musicals, and in one-man shows about George Bernard Shaw and Gilbert and Sullivan, and in cinema and television films, notably Ken Russell's Song of Summer.

Several explicit scenes of gore were cut in order to avoid an X rating, including: Maddy's death, who originally had a sickle jammed through her neck ; Ben's death, which showed Jason mashing his head into a bloody pulp ; Kate's demise, which showed Jason ramming her in the eye with a party horn, the box set ( DVD ) release only shows Jason ramming her in the eye but quickly cuts to another scene before revealing the blood and gore gushing from her eye ; we see Eddie's head hit the floor ; a shot of Russell's face splitting open with a large blood spurt ; Dan's original death had Jason ripping out his guts ; Amanda Shepard's death originally showed Jason stabbing her from behind, with the resulting blade going through her chest and subsequent blood hitting Dr. Crews ; Dr. Crews's death showed Jason's tree-trimming saw violently cutting into his stomach, sending a fountain of blood and guts in the air ; Melissa's original death had Jason cleaving her head in half with an axe with a close-up of her eyes still wriggling in their sockets.

The tape of the concession stand brawl shows Russell signing off from the broadcast as the wrestlers move away from the ring, followed by the sound of Russell's voice on a dark screen, as he hurriedly instructs the cameraman to turn his camera back on and turn it around in the " Crow's Nest " position they were taping from, located directly above the concession stand.

" Kripke himself doubted that there was any need to recognize rigid uses of definite descriptions, and argued that Russell's notion of scope offered all that was needed to account for such sentences.

The Tractatus, as Bertrand Russell saw it ( though it should be noted that Wittgenstein took strong exception to Russell's reading ), had been an attempt to set out a logically perfect language, building on Russell's own work.

In the foundations of mathematics, Russell's paradox ( also known as Russell's antinomy ), discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction.

In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory.

Other resolutions to Russell's paradox, more in the spirit of type theory, include the axiomatic set theories New Foundations and Scott-Potter set theory.

Russell's Paradox has shown us that naive set theory, based on an unrestricted comprehension scheme, is contradictory.

Note that there is a similarity between the construction of T and the set in Russell's paradox.

Therefore, depending on how we modify the axiom scheme of comprehension in order to avoid Russell's paradox, arguments such as the non-existence of a set of all sets may or may not remain valid.

* Russell's paradox: Does the set of all those sets that do not contain themselves contain itself?

In mathematics, Zermelo – Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes of naive set theory such as Russell's paradox.