Paradoxes

[MadMan YT]

I made this thread so that we can post Paradoxes.


What is a Paradox?
"a seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true".

List of Paradoxes

Self reference

• Barber paradox: A male barber shaves all and only those men who do not shave themselves. Does he shave himself? (Russell's popularization of his set theoretic paradox.)
  
• Bhartrhari's paradox: The thesis that there are some things which are unnameable conflicts with the notion that something is named by calling it unnameable.
  
• Berry paradox: The phrase "the first number not nameable in under ten words" appears to name it in nine words.
  
• Crocodile dilemma: If a crocodile steals a child and promises its return if the father can correctly guess exactly what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned?
  
• Paradox of the Court: A law student agrees to pay his teacher after (and only after) winning his first case. The teacher then sues the student (who has not yet won a case) for payment.
  
• Curry's paradox: "If this sentence is true, then Santa Claus exists."
  
• Epimenides paradox: A Cretan says: "All Cretans are liars". This paradox works in mainly the same way as the liar paradox.
  
• Grelling–Nelson paradox: Is the word "heterological", meaning "not applicable to itself", a heterological word? (A close relative of Russell's paradox.)
  
• Hilbert-Bernays paradox: If there was a name for a natural number that is identical to a name of the successor of that number, there would be a natural number equal to its successor.
  
• Kleene–Rosser paradox: By formulating an equivalent to Richard's paradox, untyped lambda calculus is shown to be inconsistent.
  
• Knower paradox: "This sentence is not known."
  
• 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."
  
  • Card paradox: "The next statement is true. The previous statement is false." A variant of the liar paradox in which neither of the sentences employs (direct) self-reference, instead this is a case of circular reference.
  • No-no paradox: Two sentences that each say the other is not true.
  • Pinocchio paradox: What would happen if Pinocchio said "My nose grows now"?[1]
  • Quine's paradox: "'Yields a falsehood when appended to its own quotation' yields a falsehood when appended to its own quotation." Shows that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals.
  • Yablo's paradox: An ordered infinite sequence of sentences, each of which says that all following sentences are false. While constructed to avoid self-reference, there is no consensus whether it relies on self-reference or not.
    
• Opposite Day: "It is opposite day today." Therefore, it is not opposite day, but if you say it is a normal day it would be considered a normal day, which contradicts the fact that it has previously been stated that it is an opposite day.
• Richard's paradox: We appear to be able to use simple English to define a decimal expansion in a way that is self-contradictory.
• Russell's paradox: Does the set of all those sets that do not contain themselves contain itself?
• Socratic paradox: "All I know is that I know nothing."
  Vagueness
  • Ship of Theseus: It seems like you can replace any component of a ship, and it is still the same ship. So you can replace them all, one at a time, and it is still the same ship. However, you can then take all the original pieces, and assemble them into a ship. That, too, is the same ship you began with.
    
• 
  
  See also List of Ship of Theseus examples
  
  • Sorites paradox (also known as the paradox of the heap): If you remove a single grain of sand from a heap, you still have a heap. Keep removing single grains, and the heap will disappear. Can a single grain of sand make the difference between heap and non-heap?
    Mathematics
    • All horses are the same color: A fallacious argument by induction that appears to prove that all horses are the same color.
      
    • Ant on a rubber rope: An ant crawling on a rubber rope can reach the end even when the rope stretches much faster than the ant can crawl.
      
    • 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.
      
    • 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.
      
    • Potato paradox: If you let potatoes consisting of 99% water dry so that they are 98% water, they lose 50% of their weight.
      
    • Russell's paradox: Does the set of all those sets that do not contain themselves contain itself?
      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: The set of all sets would have its own power set as a subset, therefore its cardinality would be at least as great as that of its power set. But Cantor's theorem proves that power sets are strictly greater than the sets they are constructed from. Consequently, the set of all sets would contain a subset greater than itself.
        
      • 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.
        
      • Skolem's paradox: Countably infinite models of set theory contain uncountably infinite sets.
        
      • 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.)
        
      • Supertasks may result in paradoxes such as
        
      • Benardete's paradox: Apparently, a man can be "forced to stay where he is by the mere unfulfilled intentions of the god".
        
      • Grandi's series: The sum of 1-1+1-1+1-1... can be either one, zero, or one-half.
        
      • Ross–Littlewood paradox: After alternately adding and removing balls to a vase infinitely often, how many balls remain?
        
      • Thomson's lamp: After flicking a lamp on and off infinitely often, is it on or off?
        
    • 
      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.
        
      • 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.
          
      • Arrow's paradox: Given more than two choices, no system can have all the attributes of an ideal voting system at once.
      • 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.
      • Decision-making paradox: Selecting the best decision-making method is a decision problem in itself.
      • Ellsberg paradox: People exhibit ambiguity aversion (as distinct from risk aversion), in contradiction with expected utility theory.
      • 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.
      • Fredkin's paradox: The more similar two choices are, the more time a decision-making agent spends on deciding.
      • Green paradox: Policies intending to reduce future CO2 emissions may lead to increased emissions in the present.
      • Hedgehog's dilemma: or Lover's paradox Despite goodwill, human intimacy cannot occur without substantial mutual harm.
      • Inventor's paradox: It is easier to solve a more general problem that covers the specifics of the sought-after solution.
      • 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?
      • Motivation crowding theory: Adding incentives for some behavior can sometimes backfire and actually result in less of that behavior.
      • Morton's fork: a type of false dilemma in which contradictory observations lead to the same conclusion.
      • 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 tolerance: Should one tolerate intolerance if intolerance would destroy the possibility of tolerance?
      • Paradox of voting: Also known as the Downs paradox. For a rational, self-interested voter the costs of voting will normally exceed the expected benefits, so why do people keep voting?
      • 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 behavior — even though they receive no benefit, or even suffer, from the change.
      • Prisoner's dilemma: Two people might not cooperate even if it is in both their best interests to do so.
      • Voting paradox: Also known as Condorcet's paradox and paradox of voting. A group of separately rational individuals may have preferences that are irrational in the aggregate.
      • Willpower paradox: Those who kept their minds open were more goal-directed and more motivated than those who declared their objective to themselves.
        
    • 
      Physics
      • Cool tropics paradox: A contradiction between modelled estimates of tropical temperatures during warm, ice-free periods of the Cretaceous and Eocene, and the lower temperatures that proxies suggest were present.
        
      • Irresistible force paradox: What would happen if an unstoppable force hit an immovable object?
        
      • Paradox of place: If everything that exists has a place, that place must have a place, and so on ad infinitum.
        
      • Paradox of the grain of millet: When a grain of millet falls it makes no sound, but when a thousand grains fall they do, thus many of nothing become something.
        
      • There were a lot of more but it's already too long if you are interested in all of them here is the link:https://en.wikipedia.org/wiki/List_of_paradoxes