The Banach–Tarski paradox is a theorem in set-theoretic geometry that 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 that can be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without c
In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share the same birthday. The birthday paradox is the counterintuitive fact that only 23 people are needed for that probability to exceed 50%.
In probability theory, the Borel–Kolmogorov paradox (sometimes known as Borel's paradox) is a paradox relating to conditional probability with respect to an event of probability zero (also known as a null set). It is named after Émile Borel and Andrey Kolmogorov.
The Boy or Girl paradox surrounds a set of questions in probability theory, which are also known as The Two Child Problem, Mr. Smith's Children and the Mrs.
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing 'the set of all ordinal numbers' leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Georg Cantor.
Cantor's diagonal argument (among various similar names) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some sense contain more elements than there are positive integers. Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardina
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 number.
In mathematics, cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The concept is understood through one-to-one correspondences between sets.
In decision theory, economics, and probability theory, the Dutch book arguments are a set of results showing that agents must satisfy the axioms of rational choice to avoid a kind of self-contradiction called a Dutch book. A Dutch book, sometimes also called a money pump, is a set of bets that ensures a guaranteed loss, i.e., the gambler will lose money no matter what happens.
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may nev
In mathematics, cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The concept is understood through one-to-one correspondences between sets.
The Flynn effect is the substantial and long-sustained increase in both fluid and crystallized intelligence test scores that were measured in many parts of the world over the 20th century, named after researcher James Flynn (1934–2020). When intelligence quotient (IQ) tests are initially standardized using a sample of test-takers, by convention the average of the test results is set to 100 and their standard deviatio
Galileo's paradox is a demonstration of one of the surprising properties of infinite sets. In his final scientific work, Two New Sciences, Galileo Galilei made apparently contradictory statements about the positive integers.
Hilbert's paradox of the Grand Hotel (colloquially the Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It shows that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often.
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
The median of a set of numbers is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as the “middle' value.
Meno (; Ancient Greek: Μένων, Ménōn) is a Socratic dialogue written by Plato around 385 BC, but set at an earlier date around 402 BC. Meno begins the dialogue by asking Socrates whether virtue (in Ancient Greek: ἀρετή, aretē) can be taught, acquired by practice, or comes by nature. In order to determine whether virtue is teachable or not, Socrates tells Meno that they first need to determine what virtue is.
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets. Usually Greek letters are used for ordinal number variables to help distinguish them from natural number variables.
The paradox of the pesticides is a paradox that states that applying pesticide to a pest may end up increasing the abundance of the pest or other pests if the pesticide upsets natural predator–prey dynamics in the ecosystem.
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 logical contradictions within modern axiomatic set theory.
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, along with appropriate group actions that act on some universe (of which the set in question is a subset), such that each partition can be mapped back onto the entire set using only finitely many distinct functions (or compositions thereof) to accomplish the mapping
Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics. Prior to quantum physics, it was thought that Quantum indeterminacy can be quantitatively characterized by a probability distribution on the set of outcomes of measurements of an observable.
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 the importance of distinguishing carefully between mathematics and metamathematics.
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. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions.
Quantitative structure–activity relationship (QSAR) models are regression or classification models used in the chemical and biological sciences and engineering. In QSAR regression models relate a set of 'predictor' variables (X) to the potency of the response variable (Y), while classification QSAR models relate the predictor variables to a categorical value of the response variable.
In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, geometric shapes, variables, or other sets. A set may be finite or infinite.
In mathematical logic and philosophy, 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 part of the Löwenheim–Skolem theorem; Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness.
The tritone paradox is an auditory illusion in which a sequentially played pair of Shepard tones separated by an interval of a tritone, or half octave, is heard as ascending by some people and as descending by others. Different populations tend to favor one of a limited set of different spots around the chromatic circle as central to the set of 'higher' tones.
In 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 to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers.
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-preserving affine transformation such that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann, assuming the axiom of choice.
Stephen Yablo (; born 1957) is a Canadian-born American philosopher. He is the Emeritus David W. Skinner Professor of Philosophy at the Massachusetts Institute of Technology (MIT) and taught previously at the University of Michigan, Ann Arbor.