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...

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...

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...

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...

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...

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

{\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...

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...

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...

{Z} ,n=2k\}} The set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation...

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...

mathematical study of infinite sets began with Georg Cantor (1845–1918). This provided some counterintuitive facts and paradoxes. For example, the number line...

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"...

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

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...

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...

Traveler's Paradox' Sets May Release For Disrupt Entertainment". Deadline Hollywood. Retrieved May 13, 2025. Tim Travers and the Time Traveler's Paradox at IMDb...

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

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 union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations...

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...

infinite sets or shapes. While the Greeks considered notions of infinity, they viewed it as paradoxical and imperfect (cf. Zeno's paradoxes), often associating...

V_{\alpha +1}} . Hilbert's paradox implies that no set with the above properties exists . For suppose V {\displaystyle V} was a set. Then V {\displaystyle...

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...

In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be...

mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four...

for the paradox to arise. The scope of these obstacles for English is comparable to that of Russell's paradox for mathematics founded on sets. One may...

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