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t. e. 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]
Proven to be impossible to prove or disprove within Zermelo–Fraenkel set theory with or without the axiom of choice (provided Zermelo–Fraenkel set theory is consistent, i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem. 1940, 1963 2nd: Prove that the axioms of arithmetic are ...
Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Every infinite set which can be enumerated by natural numbers is the ...
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. Given a set of elements {1, 2, …, n} , (henceforth referred to as the universe, specifying all possible elements under consideration) and a collection, referred to as S, of a given m subsets whose union equals the ...
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory — as a branch of mathematics — is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was ...
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.
Set Theory: An Introduction to Independence Proofs is a textbook and reference work in set theory by Kenneth Kunen. It starts from basic notions, including the ZFC axioms, and quickly develops combinatorial notions such as trees, Suslin's problem, , and Martin's axiom. It develops some basic model theory (rather specifically aimed at models of ...
Zermelo–Fraenkel set theory. In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the ...