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Please help improve this article by adding citations to reliable sources.This is a list of paradoxes, grouped thematically.Note that many of the listed paradoxes have a clear resolution.Quine's Classification of Paradoxes.Mathematical and statistical
2.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.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, excepting this one" ...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 blank" is printed, thereby making the page not blank.Liar paradox: "This sentence is false."Also "Is the answer to this question no?"Petronius' paradox: "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.Paradox of the Court: A law student agrees to pay his teacher after winning his first case.Russell's paradox: Does the set of all those sets that do not contain themselves contain itself?Vagueness
Ship of Theseus (a.That, too, is the same ship you started with.Sorites paradox: One grain of sand is not a heap.Similarly, one hair can't make the difference between being bald and not being bald.The Monty Hall paradox: which door do you choose?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.Voting paradox You can't have all the attributes of an ideal voting system at once.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.Inspection paradox: Why you will wait longer for that bus than you should.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.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
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.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.Statistical paradox: It is quite possible to draw wrong conclusions from correlation.This is also called a spurious relationship.Smallest number paradox describes how a rolling object should be able to attain a velocity of the smallest positive number.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.Monty Hall problem: An unintuitive consequence of conditional probability.Essentially the same as the Three Prisoners Problem.It appears that two sets of data separately support a certain hypothesis, but, when considered together, they support the opposite hypothesis.You are given two indistinguishable envelopes and you are told one contains twice as much money as the other.Forti paradox: If the ordinal numbers formed a set, it would be an ordinal number which is smaller than itself.Galileo's paradox: Though most numbers are not squares, there are no more numbers than squares.See also Cantor, 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.Skolem's paradox: Countably infinite models of set theory contain uncountably infinite sets.Littlewood paradox and Benardete's paradox.Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original.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).One such construction, a Morin surface, seen from "above".Smale's paradox: A sphere can, topologically, be turned inside out.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?Control paradox: Man can never be free of control, for to be free of control is to be controlled by oneself.Paradox of hedonism: When one pursues happiness itself, one is miserable; but, when one pursues something else, one achieves happiness.Newcomb's paradox: How do you play a game against an omniscient opponent?Chemical
SAR paradox: Exceptions to the principle that a small change in a molecule causes a small change in its chemical behavior are frequently profound.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.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.Ehrenfest paradox: On the kinematics of a rigid, rotating disk.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?Kuzmin limit which is a consequence of special relativity.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?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: When the traveling twin returns, he is younger and older than his sibling who stayed put.Philosophical
Epicurean paradox: The existence of evil seems to be incompatible with the existence of an omnipotent and caring God.Newcomb's paradox: A paradoxical game between two players, one of whom can predict the actions of the other.Grandfather paradox: You travel back in time and kill your grandfather before he meets your grandmother 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?"Liberal paradox: It is impossible to have both a commitment to "Minimal Liberty", and 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."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.The ontological paradox is closely tied to this, in which as a result of time travel, information or objects appear to have no beginning.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.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.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, Solow computer paradox Worker productivity may go down, despite technological improvements...Petersburg paradox: People will only offer a modest fee for a reward of infinite value.See also
Bracketing paradox
Buttered cat paradox, a humorous example of a paradox from contradicting folk tales
Ethics
Impossible object
Proof that 0.All text is available under the terms of the GNU Free Documentation License.See Copyrights for details.HOMEGAMESParadox TitlesCrusader KingsDiplomacyEuropa Universalis IIEuropa Universalis IIIEuropa Universalis III Napoleon's AmbitionEuropa Universalis: RomeHearts of IronHearts of Iron 2Hearts of Iron 2: DoomsdayHearts of Iron 2: ArmageddonVictoriaVictoria: RevolutionsOthersAnno 1701Bad Day L.Birth of AmericaBuilding WorldCity Life ed.Combat Mission: Shock ForceDawn of MagicDeath to SpiesDelware St.PARADOX MEMBERS SOCIETY DISCUSSION FORUMMeet great people, learn more about our games and get inside tips from dedicated gamers.NEWSLETTER Sign up for the Paradox newsletter, and get all the news directly form Paradox Interactive.One exhibiting inexplicable or contradictory aspects: "The silence of midnight, to speak truly, though apparently a paradox, rung in my ears" (Mary Shelley).Latin paradoxum, from Greek paradoxon, from neuter sing.Example: If your birthday is on February 29 you could state the paradox that you are thirteen years old although you have only had three birthdays.Share This
paradox A statement that seems contradictory or absurd but is actually valid or true.According to one proverbial paradox, we must sometimes be cruel in order to be kind.Another form of paradox is a statement that truly is contradictory and yet follows logically from other statements that do not seem open to objection.That which is apparently, though not actually, inconsistent with or opposed to the known facts in any case.Share This
Paradox database A relational database for Microsoft Windows, originally from Borland.Paradox 7 ran under Windows 95 and Windows NT.Share This
paradox logic An apparently sound argument leading to a contradiction.Some famous examples are Russell's paradox and the liar paradox.Web Search powered by Google
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Sitting apart from just about everything else in East Baltimore, but a set of train tracks and other warehouses is Paradox, a one time storage place for a marble company.Since November, 1991, Paradox has attracted a diversified clientele of black and white, straight and gay...Paradox was conceived after the closing of a once popular Baltimore night club, Club Fantasy; and exists to fill the void of late night dinning, dancing and entertainment all under one roof.Paradox is located in a renovated 13,000 sq.And intelligent lighting and much,
much more.The paradox arises within naive set theory by considering
the set of all sets that are not members of themselves.Such a set
appears to be a member of itself if and only if it is not a member of
itself, hence the paradox.Call the set of all sets that are not members
of themselves "R."Similarly, if
R is not a member of itself, then by definition it must be a
member of itself.Discovered by
Bertrand Russell
in 1901, the paradox has prompted much work in logic, set theory and
the philosophy and foundations of mathematics.However,
this ordinal must be both an element of the set of all ordinals and
yet greater than every such element.Forti's paradox,
Russell's paradox does not involve either ordinals or cardinals,
relying instead only on the primitive notion of set.Russell wrote to
Gottlob Frege
with news of his paradox on June 16, 1902.The paradox was of
significance to Frege's logical work since, in effect, it showed that
the axioms Frege was using to formalize his logic were
inconsistent.Specifically, Frege's Rule V, which states that two sets
are equal if and only if their corresponding functions coincide in
values for all possible arguments, requires that an expression such as
f(x) be considered both a function of the argument x
and a function of the argument f.In effect, it was this
ambiguity that allowed Russell to construct R in such a way
that it could both be and not be a member of itself.Russell's letter arrived just as the second volume of Frege's
Grundgesetze der Arithmetik (The Basic Laws of
Arithmetic, 1893, 1903) was in press.Immediately appreciating
the difficulty the paradox posed, Frege added to the
Grundgesetze a hastily composed appendix discussing
Russell's discovery.In the appendix Frege observes that the
consequences of Russell's paradox are not immediately clear.For
example, "Is it always permissible to speak of the extension of a
concept, of a class?These are the
questions," Frege notes, "raised by Mr Russell's
communication."Entitled
"Appendix B: The Doctrine of Types," the appendix represents Russell's
first detailed attempt at providing a principled method for avoiding
what was soon to become known as "Russell's paradox."Significance of the paradox
The significance of Russell's paradox can be seen once it is
realized that, using classical logic, all sentences follow from a
contradiction.Russell's paradox ultimately stems from the idea that any coherent
condition may be used to determine a set.Comprehension (or Abstraction)
axiom.This axiom in effect states that any propositional function,
P(x), containing x as a free variable can be used to
determine a set.Russell's own response to the paradox was his aptly named theory
of types.The lowest level of this hierarchy will
consist of sentences about individuals.The next lowest level
will consist of sentences about sets of sets of individuals, and so
on.In other words, before a function can
be defined, one first has to specify exactly those objects to which
the function will apply.For example, before defining the predicate
"is a prime number," one first needs to define the range of objects
that this predicate might be said to satisfy, namely the set,
N, of natural numbers.Russell's type theory thus appears in two
versions: the "simple theory" of 1903 and the "ramified theory" of
1908.Both versions have been criticized for being too ad hoc to
eliminate the paradox successfully.In addition, even if type theory
is successful in eliminating Russell's paradox, it is likely to be
ineffective at resolving other, unrelated paradoxes.Luitzen
Brouwer and the intuitionists (whose basic idea was that one cannot
assert the existence of a mathematical object unless one can also
indicate how to go about constructing it).Yet a fourth response was embodied in Ernst Zermelo's 1908
axiomatization of set theory.Zermelo's axioms were designed to
resolve Russell's paradox by again restricting the Comprehension axiom
in a manner not dissimilar to that proposed by Russell.Together, these four responses to Russell's paradox have helped
logicians develop an explicit awareness of the nature of formal
systems and of the kinds of metalogical and metamathematical results
commonly associated with them today.Bertrand Russell:
Critical Assessments, vol.Bertrand Russell: Critical Assessments, vol.Hallett, Michael (1984) Cantorian Set Theory and Limitation
of Size, Oxford: Clarendon.Russell, Bertrand (1959) My Philosophical
Development, London and New York: Routledge, 1995.Whitehead, Alfred North, and Bertrand Russell (1910, 1912, 1913)
Principia Mathematica, 3 vols, Cambridge: Cambridge
University Press.Second edition, 1925 (Vol."Your screen resolution is " + width + " x " + height + ".Free
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