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Bertrand's box paradox: A paradox of conditional probability closely related to the Boy or Girl paradox. Bertrand's paradox: Different common-sense definitions of randomness give quite different results. Birthday paradox: In a random group of only 23 people, there is a better than 50/50 chance two of them have the same birthday.
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. [ 1 ] [ 2 ] Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. [ 3 ]
The Principles of Mathematics (PoM) is a 1903 book by Bertrand Russell, in which the author presented his famous paradox and argued his thesis that mathematics and logic are identical. [ 1 ] The book presents a view of the foundations of mathematics and Meinongianism and has become a classic reference.
This category contains paradoxes in mathematics, but excluding those concerning informal logic. "Paradox" here has the sense of "unintuitive result", rather than "apparent contradiction". "Paradox" here has the sense of "unintuitive result", rather than "apparent contradiction".
With the epsilon-delta definition of limit, Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes. [35] [36] Some philosophers, however, say that Zeno's paradoxes and their variations (see Thomson's lamp) remain relevant metaphysical problems.
As Stewart Shapiro explains in his Thinking About Mathematics, Russell's attempts to solve the paradoxes led to the ramified theory of types, which, though it is highly complex and relies on the doubtful axiom of reducibility, actually manages to solve both syntactic and semantic paradoxes at the expense of rendering the logicist project ...
In one of the most important compendia of mathematical logic, compiled by Jean van Heijenoort, Richard's article is translated into English. The paradox can be interpreted as an application of Cantor's diagonal argument. It inspired Kurt Gödel and Alan Turing to their famous works.
The discovery of these paradoxes revealed that not all sets which can be described in the language of naive set theory can actually be said to exist without creating a contradiction. The 20th century saw a resolution to these paradoxes in the development of the various axiomatizations of set theories such as ZFC and NBG in common