Category: Mathematical paradoxes

Schwarz lantern
In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the
In mathematics, 0.999... (also written as 0.9, in repeating decimal notation) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal rep
Von Neumann paradox
In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preser
1 + 1 + 1 + 1 + ⋯
In mathematics, 1 + 1 + 1 + 1 + ⋯, also written , , or simply , is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1n can b
Bertrand paradox (probability)
The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work Calcul des probabilités (1889), as an example to show that the pr
Interesting number paradox
The interesting number paradox is a humorous paradox which arises from the attempt to classify every natural number as either "interesting" or "uninteresting". The paradox states that every natural nu
Newcomb's paradox
In philosophy and mathematics, Newcomb's paradox, also known as Newcomb's problem, is a thought experiment involving a game between two players, one of whom is able to predict the future. Newcomb's pa
Parrondo's paradox
Parrondo's paradox, a paradox in game theory, has been described as: A combination of losing strategies becomes a winning strategy. It is named after its creator, Juan Parrondo, who discovered the par
Hausdorff paradox
The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere (a 3-dimensional sphere in ). It states that if a certain countable subset is removed from , then
Hilbert's paradox of the Grand Hotel
Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated
Potato paradox
The potato paradox is a mathematical calculation that has a counter-intuitive result. The Universal Book of Mathematics states the problem as such: Fred brings home 100 kg of potatoes, which (being pu
Zeno's paradoxes
Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evide
The Banach–Tarski Paradox (book)
The Banach–Tarski Paradox is a book in mathematics on the Banach–Tarski paradox, the fact that a unit ball can be partitioned into a finite number of subsets and reassembled to form two unit balls. It
Braess's paradox
Braess's paradox is the observation that adding one or more roads to a road network can slow down overall traffic flow through it. The paradox was discovered by the German mathematician Dietrich Braes
Curry's paradox
Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F", requiring only a few apparently innocuous logical deduc
Painter's paradox
No description available.
Berry paradox
The Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters" (a phrase with fifty-seven letters). Bertrand Russe
Banach–Tarski paradox
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 numb
Missing square puzzle
The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only tex
Richard's paradox
In logic, 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 ordinarily used to motivate th
Sphere eversion
In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out"). Remarkably, it is possible to smoot
Girard's paradox
No description available.
Knower paradox
The knower paradox is a paradox belonging to the family of the paradoxes of self-reference (like the liar paradox). Informally, it consists in considering a sentence saying of itself that it is not kn
Paradoxes of set theory
This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logi
Staircase paradox
In mathematical analysis, the staircase paradox is a pathological example showing that limits of curves do not necessarily preserve their length. It consists of a sequence of "staircase" polygonal cha
Hilbert–Bernays paradox
The Hilbert–Bernays paradox is a distinctive paradox belonging to the family of the paradoxes of reference (like Berry's paradox). It is named after David Hilbert and Paul Bernays.
All horses are the same color
All horses are the same color is a falsidical paradox that arises from a flawed use of mathematical induction to prove the statement All horses are the same color. There is no actual contradiction, as
Cramer's paradox
In mathematics, Cramer's paradox or the Cramer–Euler paradox is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitr
String girdling Earth
String girdling Earth is a mathematical puzzle with a counterintuitive solution. In a version of this puzzle, string is tightly wrapped around the equator of a perfectly spherical Earth. If the string
Chessboard paradox
The chessboard paradox or paradox of Loyd and Schlömilch is a falsidical paradox based on an optical illusion. A chessboard or a square with a side length of 8 units is cut into four pieces. Those fou
Grandi's series
In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯, also written is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatme
Kleene–Rosser paradox
In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic are inconsistent, in particular the version of Haskell Curry's combinatory logic introduced in 19
Skolem's paradox
In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly c
Gabriel's horn
Gabriel's horn (also called Torricelli's trumpet) is a particular geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition that (albeit not strictl
Hooper's paradox
Hooper's paradox is a falsidical paradox based on an optical illusion. A geometric shape with an area of 32 units is dissected into four parts, which afterwards get assembled into a rectangle with an
1 − 2 + 3 − 4 + ⋯
In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the