Russell's paradox

English Etymology en Bertrand Russell English mathematician logician philosopher =. Proper noun en noun Russell's paradox en set theory The paradox that a set defined to contain all sets which do not contain themselves can neither consistently contain itself nor not contain itself. en 1999 R. C. Penner Discrete Mathematics: Proof Techniques and Mathematical Structures World Scientific https://books.google.com.au/books?id=t5r79vZ9ogoC&pg=PA109&dq=%22Russell%27s+paradox%22%7C%22Russell%27s+paradoxes%22&hl=en&sa=X&ved=0ahUKEwjk7t-A05HcAhWC3lQKHRmWAiQQ6AEIOzAF#v=onepage&q=%22Russell's%20paradox%22%7C%22Russell's%20paradoxes%22&f=false 109 One concludes that there must be something fishy about the Axiom of Comprehension, and, over time, the replacement of the Axiom of Comprehension by the Schema of Separation was seen to resolve Russell's paradox. Indeed, one cannot apply the Schema of Separation as in Russell's paradox unless one knows in advance that the collection of all sets is itself a set. en 2001 M. Randall Holmes Tarski's Theorem and NFU C. Anthony Anderson; Michael Zelëny Logic, Meaning and Computation: Essays in Memory of Alonzo Church Springer (Kluwer Academic) https://books.google.com.au/books?id=JIpK7Jb9s8IC&pg=PA469&dq=%22Russell%27s+paradox%22%7C%22Russell%27s+paradoxes%22&hl=en&sa=X&ved=0ahUKEwjk7t-A05HcAhWC3lQKHRmWAiQQ6AEIazAN#v=onepage&q=%22Russell's%20paradox%22%7C%22Russell's%20paradoxes%22&f=false 469 The well-known Tarski's undefinability theorem theorem of Tarski that truth of sentences in any reasonably expressive language L cannot be defined in the language L itself is proved by a diagonalization argument similar to the argument involved in Russell's paradox.It is usual to think that Russell's paradox excludes "large" sets like the universe, but this is actually not the case. An alternate solution to Russell's paradox (and other paradoxes) was proposed by Quine (1937) in his system "New Foundations" (NF): comprehension restricted to stratified formulae. 2013, Greg Frost-Arnold, Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science, Carus Publishing Company (Open Court), page 43, Roughly, the idea is that Russell's paradox reveals that certain logics suffer serious problems, and therefore these logics should be avoided.Here again, Quine asserts that the real lesson of Russell's paradox is that we should give up quantifying over abstracta. Usage notes The paradox can be stated as follows: Define . Either (a) or (b) . In case (a), ; in case (b), . In the standard axiomatisation of set theory (ZFC), the paradox is avoided by disallowing the definition of sets with criteria of Axiom schema of specification unrestricted comprehension. Synonyms paradox in set theory en Russell's antinomy Translations paradox in set theory Bulgarian: bg парадокс на Ръсел m Finnish: fi Russellin paradoksi Icelandic: is Russell-þversögn f Polish: pl paradoks Russella m See also en barber paradox en Burali-Forti paradox en Grelling-Nelson paradox Further reading Barber paradox Burali-Forti paradox Grelling-Nelson paradox Axiom schema of specification en Paradoxes en eponyms