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This is a list of paradoxes, grouped thematically. Note that many of the listed paradoxes have a clear resolution or are mistakenly identified as being a paradox for describing an ironic situation — see Quine's Classification of Paradoxes.

Logical, non-mathematical

Barbershop paradox: The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences.
Carroll's paradox: "Whatever Logic is good enough to tell me is worth writing down..."
Crocodile Dilemma: If a crocodile steals a child and promises its return if the father can correctly guess what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned?
Drinker paradox: In any pub there is a customer such that, if he or she drinks, everybody in the pub drinks.
Paradox of entailment: Inconsistent premises always make an argument valid.
Horse paradox: All horses are the same color.
Lottery paradox: It is philosophically justifiable to believe that every individual lottery ticket won't win, but not justifiable to believe that no lottery ticket will win.
Raven paradox (or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black.
Unexpected hanging paradox: The day of the hanging will be a surprise, so it cannot happen at all, so it will be a surprise. The Bottle Imp paradox uses similar logic.

Self-referential

These paradoxes have in common a contradiction arising from self-reference.

Barber paradox: An adult male barber shaves all men who do not shave themselves, and no one else. Can he shave himself? (Russell's popularisation of his set theoretic paradox.)
Berry paradox: The phrase "the first number not nameable in under ten words" appears to name it in nine words.
Curry's paradox: "If this sentence is true, the world will end in a week."
Epimenides paradox: A Cretan says: "All Cretans are liars".
Exception paradox: "If there is an exception to every rule, then every rule must have at least one exception; the exception to this one being that it has no exception." "There's always an exception to the rule, except to the exception of the rule — which is, in of itself, an accepted exception of the rule."
Grelling–Nelson paradox: Is the word "heterological", meaning "not applicable to itself," a heterological word? (Another close relative of Russell's paradox.)
Intentionally blank page: Many documents contain pages on which the text "This page is intentionally left blank" is printed, thereby making the page not blank.
Liar paradox: "This sentence is false." This is the canonical self-referential paradox. Also "Is the answer to this question no?" And "I'm lying."
Yablo's paradox: Similar to the liar paradox, but makes the self-reference less apparent by using an infinite number of sentences.
Opposite Day: "It is opposite day today."
Paradox of the Court: A law student agrees to pay his teacher after winning his first case. The teacher then sues the student (who has not yet won a case) for payment.
Petronius's Paradox: "Practice moderation in all things. Including moderation."
Quine's paradox: "Yields a falsehood when appended to its own quotation" yields a falsehood when appended to its own quotation.
Richard's paradox: We appear to be able to use simple English to define a decimal expansion in a way that is self-contradictory.
Socratic paradox: "I know that I know nothing at all."

Vagueness

Bonini's paradox: models or simulations that explain the workings of complex systems are seemingly impossible to construct: As a model of a complex system becomes more complete, it becomes less understandable; for it to be more understandable it must be less complete and therefore less accurate. When the model becomes accurate, it is just as difficult to understand as the real-world processes it represents.
Code-Talker Paradox: how can a language both enable communication and block communication?
Ship of Theseus (a.k.a. George Washington's or Grandfather's old axe): It seems like you can replace any component of a ship, and it will still be the same ship. So you can replace them all, one at a time, and it will still be the same ship. But then you can take all the original pieces, and assemble them into a ship. That, too, is the same ship with which you started.
Sorites paradox: One grain of sand is not a heap. If you don't have a heap, then adding only one grain of sand won't give you a heap. Then no number of grains of sand will make a heap. Similarly, one hair can't make the difference between being bald and not being bald. But then if you remove one hair at a time, you will never become bald. Also similar, one dollar will not make you rich, so if you keep this up, one dollar at a time, you will never become rich, no matter how much you obtain.

Mathematics

See also: Category:Mathematics paradoxes

Apportionment paradox: Some systems of apportioning representation can have unintuitive results due to rounding
- Alabama paradox: Increasing the total number of seats might shrink one block's seats.
- New states paradox: Adding a new state or voting block might increase the number of votes of another.
- Population paradox: A fast-growing state can lose votes to a slow-growing state.
Arithmetic paradoxes: Proofs of obvious contradictions; for example, "proving" that 2=1 by writing a huge expression and dividing by another expression that evaluates to zero.
Arrow's paradox/Voting paradox: Given more than two choices, no system can have all the attributes of an ideal voting system at once.
Braess's paradox: Adding extra capacity to a network can reduce overall performance.
Condorcet's paradox: A group of separately rational individuals may have preferences that are irrational in the aggregate.
Cramer's paradox: the number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve.
The infinite sum of alternating integers 1 − 2 + 3 − 4 + · · · equals $\textstyle {\frac {1}{4}}$ , which is not an integer.
Elevator paradox: Elevators can seem to be mostly going in one direction, as if they were being manufactured in the middle of the building and being disassembled on the roof and basement.
Interesting number paradox: The first number that can be considered "dull" rather than "interesting" becomes interesting because of that fact.
Intransitive dice: You can have three dice, called A, B, and C, such that A is likely to win in a roll against B, B is likely to win in a roll against C, and C is likely to win in a roll against A.
Low birth weight paradox: Low birth weight and mothers who smoke contribute to a higher mortality rate. Babies of smokers have lower average birth weight, but low birth weight babies born to smokers have a lower mortality rate than other low birth weight babies. (A special case of Simpson's paradox.)

Statistics

Accuracy paradox: a predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy.
Benford's law: In lists of numbers from many real-life sources of data, the leading digit 1 occurs much more often than the others.
Berkson's paradox: a complicating factor arising in statistical tests of proportions
It is quite possible to draw wrong conclusions from correlation. For example, towns with a larger number of churches generally have a higher crime rate — because both result from higher population. A professional organization once found that economists with a Ph.D. actually had a lower average salary than those with a BS — but this was found to be because those with a Ph.D. worked in academia, where salaries are generally lower. This is also called a spurious relationship.
Inspection paradox: Why you will wait longer for that bus than you should.
Lindley's paradox: Tiny errors in the null hypothesis are magnified when large data sets are analyzed, leading to false but highly statistically significant results
Will Rogers phenomenon: The mathematical concept of an average, whether defined as the mean or median, leads to apparently paradoxical results — for example, it is possible that moving an entry from an encyclopedia to a dictionary would increase the average entry length on both books.

Probability

Bertrand's box paradox: A paradox of conditional probability closely related to the Boy or Girl paradox.
Bertrand's paradox: Different common-sense definitions of randomness give quite different results.
Birthday paradox: What is the chance that two people in a room have the same birthday?
Borel's paradox: Conditional probability density functions are not invariant under coordinate transformations.
Boy or Girl: A two-child family has at least one boy. What is the probability that it has a girl?
False positive paradox: A test that is accurate the vast majority of the time could show you have a disease, but the probability that you actually have it could still be tiny.
Monty Hall problem: An unintuitive consequence of conditional probability.
Necktie Paradox : A wager between two people seems to favour them both. Very similar in essence to the Two-envelope paradox.
Simpson's paradox: An association in sub-populations may be reversed in the population. It appears that two sets of data separately support a certain hypothesis, but, when considered together, they support the opposite hypothesis.
Sleeping Beauty problem: A probability problem that can be correctly answered as one half or one third depending on how the question is approached.
Three cards problem: When pulling a random card, how do you determine the color of the underside?
Three Prisoners problem: More information, but unchanged probability.
Two-envelope paradox: You are given two indistinguishable envelopes and you are told one contains twice as much money as the other. You may open one envelope, examine its contents, and then, without opening the other, choose which envelope to take.

Infinity and infinitesimals

Burali-Forti paradox: If the ordinal numbers formed a set, it would be an ordinal number that is smaller than itself.
Cantor's paradox: There is no greatest cardinal number.
Galileo's paradox: Though most numbers are not squares, there are no more numbers than squares. (See also Cantor's diagonal argument)
Hilbert's paradox of the Grand Hotel: If a hotel with infinitely many rooms is full, it can still take in more guests.
Russell's paradox: Does the set of all those sets that do not contain themselves contain itself?
Skolem's paradox: Countably infinite models of set theory contain uncountably infinite sets.
Supertasks can result in paradoxes such as the Ross-Littlewood paradox and Benardete's paradox.
Zeno's paradoxes: "You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on." (This is also a physical paradox.)

Geometry and topology

The Banach–Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original.

Banach–Tarski paradox: Cut a ball into 5 pieces, re-assemble the pieces to get two balls, both of equal size to the first. The von Neumann paradox is a two-dimensional analogue.
Gabriel's Horn or Torricelli's trumpet: A simple object with finite volume but infinite surface area. Also, the Mandelbrot set and various other fractals are covered by a finite shape, but have an infinite perimeter (in fact, there are no two distinct points on the boundary of the Mandelbrot set that can be reached from one another by moving a finite distance along that boundary, which also implies that in a sense you go no further if you walk "the wrong way" around the set to reach a nearby point). This can be represented by a Klein bottle
Hausdorff paradox: There exists a countable subset C of the sphere S such that S\C is equidecomposable with two copies of itself.
Coastline paradox: the perimeter of a landmass is in general ill-defined.

**Smale's paradox** states that it is possible to turn a sphere inside out in 3-space with possible self-intersections but without creating any crease. One such construction, a Morin surface, seen from "above".

Smale's paradox: A sphere can, topologically, be turned inside out.
Missing square puzzle: Two similar figures appear to have different areas while built from the same pieces.

Decision theory

Abilene paradox: People can make decisions based not on what they actually want to do, but on what they think that other people want to do, with the result that everybody decides to do something that nobody really wants to do, but only what they thought that everybody else wanted to do.
Buridan's ass: How can a rational choice be made between two outcomes of equal value?
Chainstore paradox: Even those who know better play the so-called chain store game in an irrational manner.
Fenno's paradox: The belief that people generally disapprove of the United States Congress as a whole, but support the Congressman from their own Congressional district.
Green paradox: Policies intending to reduce future CO₂ emissions may lead to increased emissions in the present.
Kavka's toxin puzzle: Can one intend to drink the non-deadly toxin, if the intention is the only thing needed to get the reward?
Morton's fork: Choosing between unpalatable alternatives.
Navigation paradox: Increased navigational precision may result in increased collision risk.
Newcomb's paradox: How do you play a game against an omniscient opponent?
Paradox of hedonism: When one pursues happiness itself, one is miserable; but, when one pursues something else, one achieves happiness.
Paradox of tolerance: Should one tolerate intolerance; if intolerance would destroy the possibility of tolerance?
Parrondo's paradox: It is possible to play two losing games alternately to eventually win.
Prevention paradox: For one person to benefit, many people have to change their behaviour — even though they receive no benefit, or even suffer, from the change.

Physics

Robert Boyle's self-flowing flask fills itself in this diagram, but perpetual motion machines cannot exist.

Braess's paradox: Adding extra capacity to a network can reduce overall performance.
Cool tropics paradox: A contradiction between modelled estimates of tropical temperatures during warm, ice-free periods of the Cretaceous and Eocene, and the colder temperatures which proxies suggested were present.
The holographic principle: The amount of information that can be stored within a given volume is not proportional to the volume but rather to the area bounding that volume.
Irresistible force paradox: What would happen if an unstoppable force hit an immovable object?
Ontological paradox: Can a time traveler send himself information with no outside source?

Astrophysics

Algol paradox: In some binaries the partners seem to have different ages, even though they're thought to have formed at the same time.
Faint young Sun paradox: The apparent contradiction between observations of liquid water early in the Earth's history and the astrophysical expectation that the output of the young sun would have been insufficient to melt ice on earth.
The GZK paradox: High-energy cosmic rays have been observed that seem to violate the Greisen-Zatsepin-Kuzmin limit, which is a consequence of special relativity.

Biology

French paradox: the observation that the French suffer a relatively low incidence of coronary heart disease, despite having a diet relatively rich in saturated fats.
Glucose paradox: The large amount of glycogen in the liver cannot be explained by its small glucose absorption.
Gray's Paradox: Despite their relatively small muscle mass, dolphins can swim at high speeds and obtain large accelerations.
Hispanic Paradox: The finding that Hispanics in the U.S. tend to have substantially better health than the average population in spite of what their aggregate socio-economic indicators predict.
Lombard's Paradox: When rising to stand from a sitting or squatting position, both the hamstrings and quadriceps contract at the same time, despite their being antagonists to each other.
Paradox of the plankton: Why are there so many different species of phytoplankton, even though competition for the same resources tends to reduce the number of species?
Temporal paradox (paleontology): When did the ancestors of birds live?

Chemistry

Levinthal paradox : The length of time in which a protein chain finds its folded state is many orders of magnitude shorter than it would be if it freely searched all possible configurations.
SAR paradox: Exceptions to the principle that a small change in a molecule causes a small change in its chemical behavior are frequently profound.

Classical mechanics

Archimedes paradox: A massive battleship can float in a few litres of water.
Aristotle's wheel paradox: Rolling joined concentrical wheels seem to trace the same difference with their circumferences, even though the circumferences are different.
Carroll's paradox: The angular momentum of a stick should be zero, but is not.
D'Alembert's paradox: An inviscid liquid produces no drag.
Denny's paradox: Surface-dwelling arthropods (such as the water strider) should not be able to propel themselves horizontally.
Elevator paradox: Even though hydrometers are used to measure fluid density, a hydrometer will not indicate changes of fluid density caused by changing atmospheric pressure.
Feynman sprinkler: Which way will a sprinkler rotate when it is submerged in a tank and made to suck in the surrounding fluid?
Painlevé paradox: Rigid-body dynamics with contact and friction is inconsistent.
Tea leaf paradox: When stirring a cup of tea, the leaves assemble in the center, even though centrifugal force pushes them outward.

Cosmology

Bentley's paradox: In a Newtonian universe, gravitation should pull all matter into a single point.
Fermi paradox: If there are, as probability would suggest, many other sentient species in the Universe, then where are they? Shouldn't their presence be obvious?
Olbers' paradox: Why is the night sky black if there is an infinity of stars?
Thermodynamic paradox: Since the universe is not infinitely old it can not be infinite in extent.

Quantum mechanics

Bell's theorem: Measured quantum particles do not satisfy mathematical probability theory.
Einstein-Podolsky-Rosen paradox: Can far away events influence each other in quantum mechanics?
Hardy's paradox: How can we make inferences about past events that we haven't observed while at the same time acknowledge that the act of observing it affects the reality we are inferring to.
Quantum LC circuit paradox: Energies stored on capacitance and inductance are not equal to the ground state energy of the quantum oscillator.
Quantum pseudo-telepathy: Two players who can not communicate accomplish tasks that seemingly require direct contact.
Schrödinger's cat paradox: A quantum paradox — Is the cat alive or dead before we look?

Relativity

Bell's spaceship paradox: concerning relativity.
Black hole information paradox: Black holes violate a commonly assumed tenet of science — that information cannot be destroyed.
Ehrenfest paradox: On the kinematics of a rigid, rotating disk.
Ladder paradox: A classic relativity problem.
Mocanu's velocity composition paradox: a paradox in special relativity.
Supplee's paradox: the buoyancy of a relativistic object (such as a bullet) appears to change when the reference frame is changed from one in which the bullet is at rest to one in which the fluid is at rest.
Twin paradox: A puzzling consequence of special relativity: a traveling person will return younger than his identical twin who stayed put.

Thermodynamics

Gibbs paradox: In an ideal gas, is entropy an extensive variable?
Loschmidt's paradox: Why is there an inevitable increase in entropy when the laws of physics are invariant under time reversal? The time reversal symmetry of physical laws appears to allow the second law of thermodynamics to be broken.
Mpemba paradox: Hot water can under certain conditions freeze faster than cold water, even though it must pass the lower temperature on the way to freezing.

Philosophy

Fitch's paradox: If all truths are knowable, then all truths must in fact be known.
Paradox of free will: If God knew how we will decide when he created us, how can there be free will?
Grandfather paradox: You travel back in time and kill your grandfather before he conceives one of your parents, which precludes your own conception and, therefore, you couldn't go back in time and kill your grandfather.
Hutton's Paradox: If asking oneself "Am I dreaming?" in a dream proves that one is, what does it prove in waking life?
Liberal paradox: "Minimal Liberty" is incompatible with Pareto optimality.
Mere addition paradox: Is a large population barely tolerably living life better than a small happy population?
Moore's paradox: "It's raining, but I don't believe that it is."
Newcomb's paradox: A paradoxical game between two players, one of whom can predict the actions of the other.
Nihilist paradox: If truth does not exist, the statement "truth does not exist" is a truth, thereby proving itself incorrect.
Omnipotence paradox: Can an omnipotent being create a rock too heavy for themself to lift?
Paradox of hedonism: In seeking happiness, one does not find happiness.
Predestination paradox: A man travels back in time to discover the cause of a famous fire. While in the building where the fire started, he accidentally knocks over a kerosene lantern and causes a fire, the same fire that would inspire him, years later, to travel back in time. The ontological paradox is closely tied to this, in which as a result of time travel, information or objects appear to have no beginning.
Problem of evil (Epicurean paradox): The existence of evil seems to be incompatible with the existence of an omnipotent, omniscient, and morally perfect God.
Zeno's paradoxes: "You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on..." (This is also a paradox of the infinite)

Economy

See also: Category:Economics paradoxes

Allais paradox: A change in a possible outcome that is shared by different alternatives affects people's choices among those alternatives, in contradiction with expected utility theory.
Arrow information paradox: To sell information you need to give it away before the sale
Bertrand paradox: Two players reaching a state of Nash equilibrium both find themselves with no profits.
Demographic-economic paradox: nations or subpopulations with higher GDP per capita are observed to have fewer children, even though a richer population can support more children.
Diamond-water paradox (or paradox of value) Water is more useful than diamonds, yet is a lot cheaper.
Downs-Thomson paradox: Increasing road capacity at the expense of investments in public transport can make overall congestion on the road worse.
Easterlin paradox: For countries with income sufficient to meet basic needs, the reported level of happiness does not correlate with national income per person.
Edgeworth paradox: With capacity constraints, there may not be an equilibrium.
Ellsberg paradox: People exhibit ambiguity aversion (as distinct from risk aversion), in contradiction with expected utility theory.
Gibson's paradox: Why were interest rates and prices correlated?
Giffen paradox: Increasing the price of bread makes poor people eat more of it.
Icarus paradox: Some businesses bring about their own downfall through their own successes.
Jevons paradox: Increases in efficiency lead to even larger increases in demand.
Leontief paradox: Some countries export labor-intensive commodities and import capital-intensive commodities, in contradiction with Heckscher-Ohlin theory.
Lucas paradox: Capital is not flowing from developed countries to developing countries despite the fact that developing countries have lower levels of capital per worker, and therefore higher returns to capital.
Mandeville's paradox: Actions which may be qualified as vicious with regard to individuals may have benefits for society as a whole.
Metzler paradox: The imposition of a tariff on imports may reduce the relative internal price of that good.
Paradox of thrift: If everyone saves more money during times of recession, then aggregate demand will fall and will in turn lower total savings in the population.
Productivity paradox (also known as Solow computer paradox): Worker productivity may go down, despite technological improvements.
St. Petersburg paradox: People will only offer a modest fee for a reward of infinite expected value.

Perception

Tritone paradox: An auditory illusion in which a sequentially played pair of Shepard tones is heard as ascending by some people and as descending by others.

@@ Line 41: / Line 41: @@
 * [[Sorites paradox]]: One grain of sand is not a heap. If you don't have a heap, then adding only one grain of sand won't give you a heap. Then no number of grains of sand will make a heap. Similarly, one hair can't make the difference between being bald and not being bald. But then if you remove one hair at a time, you will never become bald. Also similar, one dollar will not make you rich, so if you keep this up, one dollar at a time, you will never become rich, no matter how much you obtain.
-== Mathematical ==
+== Mathematics ==
 :''See also: [[:Category:Mathematics paradoxes]]''
 * [[Apportionment paradox]]: Some systems of apportioning representation can have unintuitive results due to rounding