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.

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

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.

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.

Frank P. Ramsey tried to argue that Russell's ramification of the theory of types was unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.

Russell's paradox, which shows that the notion of the set of all those sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic and set theory.

Russell's theory of descriptions is a way of logically analyzing objects in a meaningful way regardless of that object's existence.

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.

This theory of types features prominently in Whitehead and Russell's Principia Mathematica.

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.

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.

A famous example of conceptual analysis at its best is Bertrand Russell's theory of descriptions.

Thus, " The present king of France " is analyzed, according to Russell's theory of descriptions, as " There exists an individual who is currently the king of France, there is only one such individual, and that individual is bald.

The analysis of logical concepts and the machinery of formalization that is essential to Principia Mathematica ( 3 vols., 1910 – 1913 ) ( by Bertrand Russell, 1872 – 1970, and Alfred North Whitehead, 1861 – 1947 ), to Russell's theory of descriptions, to Kurt Gödel's ( 1906 – 1978 ) incompleteness theorems, and to Alfred Tarski's ( 1901 – 1983 ) theory of truth, is ultimately due to Frege.

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

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.

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 ).

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

In The Continuum Weyl developed the logic of predicative analysis using the lower levels of Bertrand Russell's ramified theory of types.

At the same time that a split was appearing between show and working Fox terriers, a further split was occurring between two different types of white terrier, both carrying Jack Russell's name.

Includes Frege's 1879 Begriffsschrift with commentary by van Heijenoort, Russell's 1908 Mathematical logic as based on the theory of types with commentary by Willard V. Quine, Zermelo's 1908 A new proof of the possibility of a well-ordering with commentary by van Heijenoort, letters to Frege from Russell and from Russel to Frege, etc.

Russell's institutional clients include retirement plans, endowments, foundations and investment plans of all types.

# His analysis substantiated Russell's views that continuous variation of stellar spectra among the various types was due to the variation of a single parameter.

Shot six or eight times the body was draped with Russell's pistol, belt, and cartridges.

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 ).

The first opportunity for the protectionist Tories under Disraeli and Stanley to take office came in 1851, when Lord John Russell's government was defeated in the House of Commons over the Ecclesiastical Titles Act 1851.

This was brought about by Palmerston's death in 1865 and Russell's retirement in 1868.

" ( Russell's Principia Mathematica, published from 1910 to 1913, does not mention Peirce ; Peirce's work was not widely known till later.

The PDP-1 was also the original hardware for playing history's first game on a minicomputer, Steve Russell's Spacewar !.

His family was growing, and he was assisting with his brother Russell's Phat Farm clothing imprint, making Run – D. M. C.

Russell's address was inaudible to most of the Coliseum ; he was ill and died just over a week later.

After Morehead was bought out and retired Waddell merged his company with Russell's, changing the name to ' Waddell & Russell '.

The use of the vibraphone in jazz was pioneered by Paul Barbarin, the drummer with Luis Russell's band.

It was then accerlerated by Lord John Russell's instructions that all lands out of town boundaries to be sold at only £ 1 per acre.

Russell's amendment was defeated in committee.

The first response was a visit by Barbour and John Henry Paton in Allegheny in March 1876 at Russell's expense to hear their arguments, and compare the conclusions that each side had made in their studies.

In 1866, Lord John Russell's Liberal administration submitted a Reform Bill to create 400, 000 new voters, but the Bill was opposed by the ' Adullamite ' Liberals for disrupting the social order, and criticised by Radicals for not conceding the secret ballot or household suffrage.

Although it was a commercial flop, the film broke stylistic ground by being more focused and less visually ambitious – and thus easier to emulate on a tight budget – than either RHPS or Ken Russell's 1975 adaptation of The Who's music and storyline from the album Tommy, or even a lower-budget affair like The Ramones ' Rock ' n ' Roll High School ( 1979 ).

Anderson agreed to become involved with the project if his character was allowed significantly more comedic leeway than Kurt Russell's character in the feature film.

Willie Nelson and Leon Russell had a number one cover version in 1979 on the country charts, it was Russell's only number one hit on the charts.