In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition entails that it is a member of itself; if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols: Russell also showed that a version of the paradox could be derived in the axiomatic system constructed by the German philosopher and mathematician Gottlob Frege, hence undermining Frege's attempt to reduce mathematics to logic and questioning the logicist programme. Two influential ways of avoiding the paradox were both proposed in 1908: Russell's own type theory and the Zermelo set theory. In particular, Zermelo's axioms restricted the unlimited comprehension principle. With the additional contributions of Abraham Fraenkel, Zermelo set theory developed into the now-standard Zermelo–Fraenkel set theory (commonly known as ZFC when including the axiom of choice). The main difference between Russell's and Zermelo's solution to the paradox is that Zermelo modified the axioms of set theory while maintaining a standard logical language, while Russell modified the logical language itself. The language of ZFC, with the help of Thoralf Skolem, turned out to be that of first-order logic. (Wikipedia).
Russell's Paradox - A Ripple in the Foundations of Mathematics
Bertrand Russell's set theory paradox on the foundations of mathematics, axiomatic set theory and the laws of logic. A celebration of Gottlob Frege. Thank you to Professor Joel David Hamkins for your help with this video. Hi! I'm Jade. Subscribe to Up and Atom for physics, math and com
From playlist Math
Set Theory (Part 2a): Russell's Paradox
Please feel free to leave comments/questions on the video below! In this video, I briefly speak about the Russell paradox and why ZFC avoids this paradox when discussing pathological sets. One should hopefully see why it is that this paradox is disastrous for the naive set theory adopted
From playlist Set Theory by Mathoma
What is Bertrand Russels Barber Paradox?
Logical paradoxes are some of the most infuriating and frustrating problems that we can try to solve. As humans, we always want to find an answer and we naturally assume that an answer must exist. In the case of Bertrand Russell's Barber Paradox, a solution does exist, but it becomes even
From playlist Concerning Questions
Bertrand Russell's Barber Paradox
SUPPORT CR on PATREON: http://bit.ly/2qBHcvf ADDITIONAL ASSUMPTIONS: 1. Everyone is male. 2. The townspeople can only shave themselves or be shaved by the barber, they cannot shave anyone else. 3. The barber must be shaved. Logical paradoxes are some of the most infuriating and frustr
From playlist Concerning Everything
Metaphilosophy & Russell vs Wittgenstein on Judgment
After a brief discussion of the nature of philosophy and the origins of analytic philosophy, Fraser MacBride discusses Bertrand Russell and Ludwig Wittgenstein on the nature of relations and the structure of judgment or belief, how our thought relates to external objects in the world. Amon
From playlist Wittgenstein
Physics - Relativity: Understanding Space, Time & Relativity (11 of 55) What is the Twin Paradox?
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the twin paradox. Time paradox is the time experienced by observer B (on space ship at v=0.99c) is different compare to the time experienced by observer A (on Earth). If the voyage takes 70 ye
From playlist PHYSICS 62.1 UNDERSTANDING RELATIVITY & SPACE
Most paradoxes either stem from the misunderstanding of a topic, or aren't really paradoxes. However, here is a paradox that seems to contradict logic itself. What's going on here? And what does the liar paradox have to do with computer science? #some2
From playlist Summer of Math Exposition 2 videos
The Andromeda Paradox Explained: Loss of simultaneity in special relativity
#SpecialRelativity Bob is sitting on a park bench feeding pigeons. He sees Alice heading in his direction, enjoying a morning run. “How are you?” he calls. Breathless, she responds. “Fine. But an armada from the Andromeda Galaxy has just departed for Earth.” Bob laughs. “No, it hasn’t!” Wh
From playlist Special Relativity - A Gentle Introduction
Relativity: how people get time dilation wrong
Einstein’s special theory of relativity is notorious for being easy to misuse, with the result that sometimes result in claims of paradoxes. When one digs more carefully into the theory, you find that no such paradoxes actually exist. In this video, Fermilab’s Dr. Don Lincoln describes a
From playlist Relativity
Barber & Russell Paradoxes (History of Undecidability Part 2) - Computerphile
$20 off your 1st purchase at www.LittleBits.com use the code “COMPUTERPHILE” The Barber Paradox: Professor Brailsford continues the history of undecidability. History of Undecidability Part1: http://youtu.be/nsZsd5qtbo4 Turing & The Halting Problem: http://youtu.be/macM_MtS_w4 http://
From playlist The History of Undecidability
Why it took 379 pages to prove 1+1=2
Sign up to Brilliant to receive a 20% discount with this link! https://brilliant.org/upandatom/ Hi! I'm Jade. If you'd like to consider supporting Up and Atom, head over to my Patreon page :) https://www.patreon.com/upandatom Visit the Up and Atom store https://store.nebula.app/collecti
From playlist Math
1.11.9 Russell's Paradox: Video
MIT 6.042J Mathematics for Computer Science, Spring 2015 View the complete course: http://ocw.mit.edu/6-042JS15 Instructor: Albert R. Meyer License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.042J Mathematics for Computer Science, Spring 2015
Logic: The Structure of Reason
As a tool for characterizing rational thought, logic cuts across many philosophical disciplines and lies at the core of mathematics and computer science. Drawing on Aristotle’s Organon, Russell’s Principia Mathematica, and other central works, this program tracks the evolution of logic, be
From playlist Logic & Philosophy of Mathematics
Turing Meets Paradoxes (History of Undecidability Part 3) - Computerphile
Taking a solemn oath to promise never to write a program that analyses other programs? - That's how Professor Brailsford felt when he first understood undecidability. Latest in our series on the History of Undecidability Undecidability Tangent (History of Undecidability Part 1): http://y
From playlist The History of Undecidability
Bertrand Russell on Hegel (1957)
A few clips of Bertrand Russell discussing Hegel and his journey away from the Hegelian commitments of his early years. #Philosophy #BertrandRussell #Hegel
From playlist Shorter Clips & Videos - Philosophy Overdose
You should know what Impredicativity is.
In this video I discuss the concept of predicativity, impredicativity and vicious circles. The text for the video is found in https://gist.github.com/Nikolaj-K/aae1f4bd582e60e6b7e5b5431fee054c
From playlist Logic