In set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is two families of disjoint subsets...

In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician...

be self referential without being paradoxical ("This statement is written in English" is a true and non-paradoxical self-referential statement), self-reference...

well order any set allows certain constructions to be performed that have been called paradoxical. One example is the Banach–Tarski paradox, a theorem widely...

construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions...

discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were...

{\displaystyle x\notin x} , it would state the existence of Russell's paradoxical set, giving a contradiction. It was this contradiction that led the axiom...

other assumption is also disproved" leads to paradoxical consequences. Not to be confused with the Barber paradox. What the Tortoise Said to Achilles: If a...

as a way to have set-like collections while differing from sets so as to avoid paradoxes, especially Russell's paradox (see § Paradoxes). The precise definition...

axiomatic set theory is free from paradox. Algebra of sets Alternative set theory Category of sets Class (set theory) Family of sets Fuzzy set Mereology...

century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial...

This is the paradoxical nature of Cantor's "paradox". While Cantor is usually credited with first identifying this property of cardinal sets, some mathematicians...

to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and...

without restriction, leads to paradoxes. One common example is Russell's paradox: there is no set consisting of "all sets that do not contain themselves"...

from the point of view of M, was seen as paradoxical in the early days of set theory; see Skolem's paradox for more. The minimal standard model includes...

the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories...

In set theory, the complement of a set A, often denoted by A ∁ {\displaystyle A^{\complement }} (or A′), is the set of elements not in A. When all elements...

Russell's paradox, the result is typically called a paradox and was described as a "paradoxical state of affairs" by Skolem. In model theory, a model...

other paradoxes listed are difficult to interpret. "What the Tortoise Said to Achilles", written in 1895 by Lewis Carroll, describes a paradoxical infinite...

between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships...

In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing...

the rn (thus, r is an undefinable number). This is the paradoxical contradiction. Richard's paradox results in an untenable contradiction, which must be...

mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle...

it to eliminate the very principle upon which that freedom relies is paradoxical. Michel Rosenfeld, in the Harvard Law Review in 1987, stated: "it seems...

contradict themselves Banach–Tarski paradox – Geometric theorem Galileo's paradox – Paradox in set theory Paradoxes of set theory Pigeonhole principle – If...

mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed...

association with gods or beliefs. Instead, it was understood as a paradoxical set of reciprocal ideas: voluntary will and voluntary restraint in the...

mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related...

the source of contradictions like Russell's paradox and Richard's paradox.: 27  Cantor considered the set T of all infinite sequences of binary digits...

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence...