List of paradoxes

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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.

[edit] Logical, non-mathematical

• Paradox of entailment: Inconsistent premises always make an argument valid.
• Raven paradox (or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black.
• Horse paradox: All horses are the same color.
• 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.
• Drinker paradox: In any pub there is a customer such that, if he or she drinks, everybody in the pub drinks.
• Carroll's paradox: "Whatever Logic is good enough to tell me is worth writing down..."
• 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.

[edit] Self-referential

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

• Barber paradox: The adult male barber who 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."
• 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?"
• 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 which is self-contradictory.
• Opposite Day: It is opposite day today.

[edit] Vagueness

• 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, or 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.

[edit] Mathematical

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 which are irrational in the aggregate.
• 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.)
• Missing dollar paradox: Faulty logic makes it appear as if a dollar from a restaurant bill has gone missing. Not in the same class as the others.
• Smallest number paradox describes how a rolling object should be able to attain a velocity of the smallest positive number.

[edit] Statistics

• 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.

[edit] Probability

The Monty Hall paradox: which door do you choose?

See also: Category:Probability theory paradoxes

• Bertrand's paradox (probability): 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?
• 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?
• 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.

[edit] Infinity and infinitesimals

• Burali-Forti paradox: If the ordinal numbers formed a set, it would be an ordinal number which 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.)

[edit] 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.
• 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.

[edit] Decision theoretic

• 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?
• 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.
• 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.
• Rahel's paradox: Every model of reality might be missing a crucial ingredient, except the perfect model. The perfect model, however, is useless because it is already given by reality. Therefore, any model is useless.

[edit] Chemical

• 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.

[edit] Physical

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

• Archimedes paradox: A massive battleship can float in a few litres of water.
• Bell's spaceship paradox: concerning relativity.
• Bell's theorem: Measured quantum particles do not satisfy mathematical probability theory.
• Black hole information paradox: Black holes violate a commonly assumed tenet of science — that information cannot be destroyed.
• Braess' paradox: Sometimes adding extra capacity to a network can reduce overall performance.
• 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.
• Ehrenfest paradox: On the kinematics of a rigid, rotating disk.
• Einstein-Podolsky-Rosen paradox: Can far away events influence each other in quantum mechanics?
• 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?
• Gibbs paradox: In an ideal gas, is entropy an extensive variable?
• The GZK paradox: High-energy cosmic rays have been observed which seem to violate the Greisen-Zatsepin-Kuzmin limit which is a consequence of special relativity.
• 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.
• The Irresistible force paradox: what would happen if an unstoppable force hits an immovable object?
• Ladder paradox: A classic relativity problem.
• 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.
• Olbers' paradox: Why is the night sky black if there is an infinity of stars?
• Ontological paradox: Can a time traveler send himself information with no outside source?
• 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?
• 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.
• Temporal paradox (paleontology): When did the ancestors of birds live?
• Twin paradox: A puzzling consequence of special relativity: a traveling person will return younger than his identical twin who stayed put.
• 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.

[edit] Philosophical

• Epicurean paradox: The existence of evil seems to be incompatible with the existence of an omnipotent and caring God.
• Fitch's paradox: If all truths are knowable, then all truths must in fact be known.
• 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 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.
• 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)

[edit] Economic

See also: Category:Economics paradoxes

• Abilene paradox: A group of people often has to decide against each member's own personal interests or views.
• Allais paradox: A change in a possible outcome which is shared by different alternatives affects people's choices among those alternatives, in contradiction with expected utility theory.
• Bertrand paradox: Two players reaching a state of Nash equilibrium both find themselves with no profits.
• Diamond-water paradox (or paradox of value) Water is more useful than diamonds, yet is a lot cheaper.
• 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.
• 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.
• 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.
• Parrondo's paradox: It is possible to play two losing games alternately to eventually win.
• 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.

[edit] See also

• Bracketing paradox
• Buttered cat paradox, a humorous example of a paradox from contradicting folk tales
• Ethics
• Impossible object
• Proof that 0.999... equals 1
• Logical fallacy
• Puzzle