Russell's paradox

Is God a Mathematician?

by Mario Livio  · 6 Jan 2009  · 315pp  · 93,628 words

noted: “More than once in history the discovery of paradox has been the occasion for major reconstruction at the foundation of thought.” Russell’s paradox provided for precisely such an occasion. Russell’s Paradox The person who essentially single-handedly founded the theory of sets was the German mathematician Georg Cantor. Sets, or classes, quickly proved

The Information: A History, a Theory, a Flood

by James Gleick  · 1 Mar 2011  · 855pp  · 178,507 words

set this way: S is the set of all sets that are not members of themselves. This version is known as Russell’s paradox. It cannot be dismissed as noise. To eliminate Russell’s paradox Russell took drastic measures. The enabling factor seemed to be the peculiar recursion within the offending statement: the idea of sets

Journey to the Edge of Reason: The Life of Kurt Gödel

by Stephen Budiansky  · 10 May 2021  · 406pp  · 108,266 words

is not, then by definition it should be included. But if it is included, that means it is a member of itself, so should not. “Russell’s Paradox,” as it came to be known, echoed paradoxes that had been around since antiquity. The prototype is the Liar’s Paradox, attributed to Epimenides the

Fermat’s Last Theorem

by Simon Singh  · 1 Jan 1997  · 289pp  · 85,315 words

not teaspoons is one of the things that are not teaspoons.’ It was this curious and apparently innocuous observation that led to the catastrophic paradox. Russell’s paradox is often explained using the tale of the meticulous librarian. One day, while wandering between the shelves, the librarian discovers a collection of catalogues. There

The Age of Spiritual Machines: When Computers Exceed Human Intelligence

by Ray Kurzweil  · 31 Dec 1998  · 696pp  · 143,736 words

, and/or precision. Robotics The science and technology of designing and manufacturing robots. Robotics combines artificial intelligence and mechanical engineering. ROM See Read-Only Memory. Russell’s Paradox The ambiguity created by the following question: Does a set that is defined as “all sets that do not include themselves” include itself as a

The Master and His Emissary: The Divided Brain and the Making of the Western World

by Iain McGilchrist  · 8 Oct 2012

, is like the Achilles paradox: with the loss of unity, the attempt is made to recapture it by summing parts, but necessarily fails. 7. Cf. Russell’s paradox: the set of all sets that are not members of themselves leads inexorably to a contradiction. If such a set were a member of itself

The Man From the Future: The Visionary Life of John Von Neumann

by Ananyo Bhattacharya  · 6 Oct 2021  · 476pp  · 121,460 words

there are infinitely many. Mathematicians had embraced Cantor’s theory as a powerful tool for manipulating and proving theorems about such sets of infinite size. Russell’s paradox, however, threatened to deal a far more serious blow to set theory than earlier ideological objections. The problem was this: consider a set of objects

Types and Programming Languages

by Benjamin C. Pierce  · 4 Jan 2002  · 647pp  · 43,757 words

quantifiers. The terms "predicative" and "impredicative" originate in logic. Quine (1987) offers a lucid summary of their history: In exchanges with Henri Poincaré...Russell attributed [Russell's] paradox tentatively to what he called a vicious-circle fallacy. The "fallacy" consisted in specifying a class by a membership condition that makes reference directly or

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indirectly to a range of classes one of which is the very class that is being specified. For instance the membership condition behind Russell's Paradox is non-self-membership: x not a member of x. The paradox comes of letting the x of the membership condition be, among other things

The Haskell Road to Logic, Maths and Programming

by Kees Doets, Jan van Eijck and Jan Eijck  · 15 Jan 2004

)? 3. How many pairs of curly braces occur in the expanded notation for ℘ 5 (∅), in the representation where ∅ appears as 0? 4.10 Further Reading Russell’s paradox, as stated by Russell himself, can be found in [Rus67]. A further introduction to sets and set theory is Doets, Van Dalen and De Swart

Infinite Ascent: A Short History of Mathematics

by David Berlinski  · 2 Jan 2005  · 158pp  · 49,168 words

attributes of a myth. Cantor’s set theory is inconsistent, mathematicians understanding at once that its fecundity and its inconsistency were deeply linked. There is Russell’s paradox, the best known among a collection of paradoxes and the easiest to state. If sets are subject only to some principle of free construction, then

Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers

by Amir D. Aczel  · 6 Jan 2015  · 204pp  · 60,319 words

Seven Databases in Seven Weeks: A Guide to Modern Databases and the NoSQL Movement

by Eric Redmond, Jim Wilson and Jim R. Wilson  · 7 May 2012  · 713pp  · 93,944 words

The Demon in the Machine: How Hidden Webs of Information Are Finally Solving the Mystery of Life

by Paul Davies  · 31 Jan 2019  · 253pp  · 83,473 words

The Infinite Book: A Short Guide to the Boundless, Timeless and Endless

by John D. Barrow  · 1 Aug 2005  · 292pp  · 88,319 words