This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive...

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

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

set theory is inconsistent. Prior to Russell's paradox (and to other similar paradoxes discovered around the time, such as the Burali-Forti paradox)...

several paradoxes—presumably had in mind. Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining...

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

In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal...

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...

Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is...

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

in the development of modern logic and set theory. Thought experiments can also yield interesting paradoxes. The grandfather paradox, for example, would...

Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness...

"Set Theory", Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University Suppes, Patrick (1972) [1960], Axiomatic Set Theory,...

set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF)...

Skolem's paradox is the apparent contradiction that a countable model of first-order set theory could contain an uncountable set. The paradox arises from...

of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set...

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...

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction...

In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, κ i {\displaystyle \kappa _{i}} and λ i {\displaystyle \lambda _{i}}...

This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list...

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary...

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...

included as one of its members). This paradox prevents the existence of a universal set in set theories that include either Zermelo's axiom of restricted comprehension...

set theory (KP), pronounced /ˈkrɪpki ˈplɑːtɛk/, is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as...

paradox Liar paradox List of paradoxes Richard's paradox Zermelo–Fraenkel set theory Curry, Haskell B. (Sep 1942). "The Inconsistency of Certain Formal...

philosophy of mathematics NP (complexity) – Complexity class used to classify decision problems Paradoxes of set theory Transcomputational problem – Class of computational...

of the original paradoxes that added to the need for a formalized set theory to avoid these paradoxes. This paradox is usually resolved in formal set...

sentences of a given language cannot be defined within that language. To formulate linguistic theories without semantic paradoxes such as the liar paradox, it...

condition, leading to paradoxes such as Russell's paradox in naïve set theory. naive set theory 1.  Naive set theory can mean set theory developed non-rigorously...

Less well known [than other paradoxes] is the paradox of tolerance: Unlimited tolerance must lead to the disappearance of tolerance. If we extend unlimited...