Achilles and the tortoise paradox | arrow paradox | Cantor's paradox | dichotomy paradox | Epimenides paradox | Eubulides paradox | Hilbert hotel paradox | Monty Hall problem | Russell's antinomy | Saint Petersburg paradox | Socrates' paradox | stade paradox | two envelopes paradox (total: 13)
The Achilles and the tortoise paradox is the conundrum that fleet-of-foot Achilles apparently cannot catch a plodding tortoise which has been given a head start, since during the time it takes Achilles to catch up to a given position, the tortoise has moved forward some distance. The conclusion is clearly fallacious since Achilles obviously will pass the tortoise.
Consider an arrow in flight, which has an instantaneous position at a given instant of time. At such an instant, it is indistinguishable from a motionless arrow in the same position. Despite this fact, the motion of the arrow nonetheless occurs and is perceived, which is a conundrum known as the arrow paradox.
Cantor's paradox is the logical conundrum that the set of all sets is its own power set. Therefore, the cardinality of the set of all sets must be bigger than itself.
The dichotomy paradox is the conundrum that before an object can travel a given distance d, it must travel a distance d/2. But in order to travel a distance d/2, it must travel a distance d/4. And so on. Since this sequence goes on forever, it appears that the distance d cannot be traveled.
The Epimenides paradox is the self-contradiction inherent in the statement, "All Cretans are liars; one of their own poets has said so."
The Eubulides paradox is the conundrum posed by the self-contradiction of the statement, "This statement is false."
Let a hotel have a denumerable set of rooms numbered 1, 2, 3, .... Then the Hilbert hotel paradox is the conundrum that any finite number n of guests can be accommodated without evicting the current guests by moving the current guests from room i to room i + n. Furthermore, a denumerable number of guests can be similarly accommodated by moving the existing guests from i to 2i, freeing up the denumerable number of rooms 2i - 1.
The Monty Hall problem considers three closed doors, behind one of which is a car and behind the other two of which are goats, you are asked to pick a door and will win whatever is behind it. But before that door is opened, someone who knows what's behind the doors opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door, that is, the door which neither you picked nor he opened.
Let R be the set of all sets which are not members of themselves. Then Russell's antinomy is the conundrum that R is neither a member of itself nor not a member of itself.
Consider a hypothetical game in which a player bets on how many tosses of a coin will be needed before it first turns up heads. The player pays a fixed amount initially, and then receives 2^n dollars if the coin comes up heads on the nth toss. The expectation value of the gain is then 1/2 × 2 + 1/4 × 4 + 1/8 × 8 + ... = 1 + 1 + 1 + ... = ∞ dollars. The Saint Petersburg paradox is that in this game, any finite amount of money can be wagered and the player will still come out ahead on average.
| solution Achilles and the tortoise paradox | The resolution of the paradox awaited calculus and the proof that infinite geometric series can converge, so that the infinite number of catch-ups needed is balanced by the increasingly short amount of time needed to traverse the distances. arrow paradox | The resolution of the paradox awaited calculus and the rigorous treatment of infinitesimal quantities. dichotomy paradox | The resolution of the paradox awaited calculus and the proof that infinite geometric series such as sum_(i=1)^∞(1/2)^i = 1 can converge, so that the infinite number of "half-steps" needed is balanced by the increasingly short amount of time needed to traverse the distances. Monty Hall problem | Yes. (Assuming you prefer to win a car rather than a goat.) stade paradox | The resolution of the paradox awaited calculus and the rigorous treatment of infinitesimal quantities. two envelopes paradox | The probabilistic argument is fallacious. There is a 50% chance of increasing the amount and 50% chance of decreasing it by switching.
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.
That which is in locomotion must arrive at the half-way stage before it arrives at the goal.
Epimenides was a Cretan who made one immortal statement: "All Cretans are liars." One of Crete's own prophets has said it: "Cretans are always liars, evil brutes, lazy gluttons." He has surely told the truth. For this reason correct them sternly, that they may be sound in faith instead of paying attention to Jewish fables and to commandments of people who turn their backs on the truth.
This sentence is false. The next sentence is false. The previous sentence is true.
For any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers.
Let R = {x:x not element x}. Then R element R iff R not element R.
| formulation date | formulators | status Achilles and the tortoise paradox | 460 BC (2484 years ago) | Zeno of Elea | resolved arrow paradox | 460 BC (2484 years ago) | Zeno of Elea | resolved Cantor's paradox | 1899 (126 years ago) | Georg Cantor | proved dichotomy paradox | 460 BC (2484 years ago) | Zeno of Elea | resolved Epimenides paradox | 550 BC (2574 years ago) | Epimenides | not necessarily a true paradox Eubulides paradox | 350 BC (2374 years ago) | Eubulides | paradox Hilbert hotel paradox | | David Hilbert | paradox Monty Hall problem | | | proved in the affirmative Russell's antinomy | 1901 (124 years ago) | Bertrand Russell | paradox Saint Petersburg paradox | 1713 (312 years ago) | Nicolaus I Bernoulli | Socrates' paradox | 430 BC (2454 years ago) | Socrates | paradox stade paradox | 460 BC (2484 years ago) | Zeno of Elea | resolved two envelopes paradox | 1953 (72 years ago) | Maurice Kraitchik | fallacy | proof date | provers Saint Petersburg paradox | 1738 (25 years later) (287 years ago) | Daniel Bernoulli | additional people involved | associated entities Achilles and the tortoise paradox | Achilles | Aristotle | Simplicius Of Cilicia | Achilles arrow paradox | Aristotle | Simplicius Of Cilicia | (none) dichotomy paradox | Aristotle | Simplicius Of Cilicia | (none) Epimenides paradox | Bertrand Russell | (none) Russell's antinomy | Ernst Friedrich Ferdinand Zermelo | (none) stade paradox | Aristotle | Simplicius Of Cilicia | (none) two envelopes paradox | Martin Gardner | (none)
R = {x:x not element x}
Not a true paradox since the poet may have knowledge that at least one Cretan is, in fact, honest, and so is lying when he says that all Cretans are liars. There therefore need be no self-contradiction in what could simply be a false statement by a person who is himself a liar.
The statement is paradoxical because there is no way to assign a consistent classical binary truth value to it.
The statement is provably true; it is a paradox only because it is counterintuitive. The counterintuitive nature derives from the fact that for infinitely many rooms, the two statements "every room is occupied" and "no more guests can be accommodated" are not equivalent.
The paradox can be resolved by refining the decision model using the concept of marginal utility. The paradox can be resolved by taking into account the finite resources of the participants. The paradox can be resolved by noting that one cannot buy that which is not sold.
mathematical paradoxes
