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In mathematical logic, Russell's paradox 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.
Saul Aaron Kripke is an American philosopher and logician in the analytic tradition. He is a Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emeritus professor at Princeton University. Since the 1960s, Kripke has been a central figure in a number of fields related to mathematical logic, modal logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology, and recursion theory. Much of his work remains unpublished or exists only as tape recordings and privately circulated manuscripts.
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The axiom of reducibility was introduced by Bertrand Russell in the early 20th century as part of his ramified theory of types. Russell devised and introduced the axiom in an attempt to manage the contradictions he had discovered in his analysis of set theory.
The mathematical concept of a function emerged in the 17th century in connection with the development of the calculus; for example, the slope of a graph at a point was regarded as a function of the x-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme.
The type theory was initially created to avoid paradoxes in a variety of formal logics and rewrite systems. Later, type theory referred to a class of formal systems, some of which can serve as alternatives to naive set theory as a foundation for all mathematics.
Predication in philosophy refers to an act of judgement where one term is subsumed under another. A comprehensive conceptualization describes it as the understanding of the relation expressed by a predicative structure primordially through the opposition between particular and general or the one and the many.
References
- ↑ Yaqub, Aladdin M. (2013), An Introduction to Logical Theory, Broadview Press, p. 288, ISBN 9781551119939 .
- ↑ Oppy, Graham (2007), Ontological Arguments and Belief in God, Cambridge University Press, p. 145, ISBN 9780521039000 .
- ↑ Kremer, Michael (1985), "Frege's theory of number and the distinction between function and object", Philosophical Studies, 47 (3): 313–323, doi:10.1007/BF00355206, MR 0788101 .
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