Start hereA fast runner can never catch a tortoise. A sentence can be true only if it is false. Both claims sound absurd, yet each one comes from reasoning that looks fine at every step. So which step goes wrong?
A paradox is an argument that starts from ideas that seem true, moves by steps that seem fine, and still lands somewhere that seems false. The Stanford Encyclopedia of Philosophy puts it as “apparently impeccable reasoning from apparently impeccable premises” that “yields a falsehood”.
Below are 12 famous ones in four groups. Each card says what the puzzle is, what the sources say, and how settled it is.
Puzzles from ancient Greece
Four of the oldest puzzles come from Greek philosophers, and people still discuss how best to answer some of them.
1. Achilles and the tortoise
A fast runner seems unable to catch a slow tortoise that starts a little ahead.
Scholars estimate that Zeno of Elea was born around 490 BC, and his book of paradoxes has not survived. What we know of his arguments comes second-hand, mainly through Aristotle.
The puzzle goes like this. Before Achilles can catch the tortoise, he must reach the point where it started. By then it has crawled a little further, and this repeats without end. So Zeno concludes that Achilles never catches it.
The usual modern answer is that an endless list of shorter and shorter distances can still add up to a finite total. The encyclopedia calls this mathematical approach the “received” view. It adds that a complete answer needs a rigorous account of adding up infinitely many terms.
For example, walk half the way to a door, then half of what is left, then half again. The steps never run out on paper, yet you reach the door.
How sure: Strong the encyclopedia calls the mathematical answer the received view, with some qualifications, and that answer uses mathematics worked out in the 1800s.
2. The flying arrow
An arrow in flight seems to stand still at every single moment, so how can it move?
As Aristotle reports it, Zeno argued that the flying arrow is at rest, if time is made of moments. At any one instant, the arrow travels no distance. If time is nothing but instants, it seems the arrow never moves.
One modern reply is that speed only makes sense over a stretch of time. Speed is distance divided by time, and at a single instant that is 0 divided by 0, which is not a number at all. On this view, motion happens only over stretches of time, which the encyclopedia calls counter-intuitive.
For example, one frame of a film shows a runner frozen mid-step. You only see the movement across many frames.
How sure: Strong the encyclopedia calls this answer correct, while noting that it feels strange.
3. The liar
“This sentence is false” seems to be true if it is false, and false if it is true.
The encyclopedia’s own example is a sentence that calls itself a lie: if it is true, it is a lie and so not true. But if it is not true, then what it says is true after all.
It was discussed in classical times by the Megarians, and also by Aristotle and Cicero. It is sometimes called the Epimenides paradox, after a Cretan said to have claimed that all Cretans are always liars.
The puzzle seems to follow from logic plus a few obvious ideas about truth. That is why logicians take it seriously. Work on it became part of the development of modern mathematical logic.
For example, picture a notice board with one card on it that reads “Every notice on this board is untrue”.
How sure: Unsettled thousands of years have produced a great number of proposed solutions, too many for the encyclopedia to examine them all.
4. The heap
One grain of sand is not a heap, and one more grain never seems to make the difference, so when does a heap start?
The Megarian philosopher Eubulides, in the 4th century BC, is usually credited with the first version. The name “sorites” comes from the Greek word soros, meaning “heap”.
The trouble comes from vague terms: words with blurry edges, such as “bald”, “tall” or “old”. Because “heap” has unclear boundaries, no single grain seems to make the difference, which leads to the absurd conclusion that no number of grains makes a heap. Run backwards, the same reasoning suggests that even a single grain makes a heap.
The philosopher Dorothy Edgington points to everyday versions, such as a chore where it never seems to matter whether it is done today or tomorrow. For example, one chocolate makes no real difference to a diet, and neither does the next one. Small steps that never seem to matter can still add up.
How sure: Unsettled most theorists of vagueness think the argument has a flaw that can be found, and the encyclopedia sets out several rival ways to answer it.
Puzzles of sets, infinity and knowing
Three more puzzles look like word games, and each one has drawn serious work from mathematicians or philosophers.
5. Russell’s paradox and the barber
Think of the set of all sets that are not members of themselves: is it a member of itself?
A set is simply a collection, such as the set of all teacups. If this odd set is a member of itself, it must not be, and if it is not, then it must be.
Bertrand Russell wrote to Gottlob Frege with news of the paradox on 16 June 1902. Frege responded with both dismay and admiration. The problem struck at his major work, the Basic Laws of Arithmetic (Grundgesetze der Arithmetik). Russell’s own answer was his theory of types, and the paradox prompted major advances in logic, set theory and the foundations of mathematics.
The barber is a well-known cousin.
A village barber shaves all and only the villagers who do not shave themselves. Who shaves the barber? The logician W. V. O. Quine answered that there is an overwhelming presumption that such a class exists, but no presumption that such a barber does. Some later writers argue the two are closer than Quine allowed.
For example, a library catalogue that lists every catalogue not listing itself runs into the same knot.
How sure: Strong the contradiction itself is not in doubt, and set theories such as Zermelo’s offer standard ways around it.
6. Galileo’s paradox of infinity
There seem to be fewer square numbers than whole numbers, yet the two can be paired off one to one.
In Two New Sciences (1638), Galileo set out the puzzle: the squares are only some of the whole numbers, yet each whole number can be matched with exactly one square. Pair 1 with 1, 2 with 4, 3 with 9, and so on. Neither list ever runs out.
Galileo concluded that “equal”, “greater” and “smaller” cannot be applied to infinite collections. Modern set theory, following Cantor, rejects his conclusion: two collections that can be paired off one to one count as the same size.
So in today’s mathematics, part of an infinite collection can be as large as the whole. For example, the even numbers can be paired off with all the whole numbers too, by matching each number with its double.
How sure: Strong it is the standard view in mathematics, though it gives up the idea that a part is always smaller than the whole.
7. The surprise test
A teacher announces a surprise test next week, and a student “proves” it cannot happen.
The class meets on Monday, Wednesday and Friday. The student argues that a Friday test would be predictable on Thursday, so it would not be a surprise. The same reasoning then rules out Wednesday, and then Monday. Yet common sense says surprise tests are possible even after a warning.
The first published reaction, by D. J. O’Connor in 1948, sided with the student and called the announcement self-defeating. In 1953 the logician W. V. O. Quine argued that the announcement does not give the student knowledge that a surprise test will come.
For example, a friend says, “I’ll drop by one day this week as a surprise.” A visit on Wednesday could still catch you off guard.
How sure: Unsettled the encyclopedia says riddles like this one typically have conflicting answers, each with good credentials.
Puzzles of chance and choice
Three puzzles show how working out the best bet can clash with common sense.
8. The St. Petersburg game
A coin game seems worth any price to play, yet paying a large sum for it seems absurd.
To value a bet, decision theorists multiply each prize by its chance and add up the results. That total is called the expected value. Taken at face value, the theory says it would be rational to pay any finite fee to play once, although you will almost certainly win a modest amount.
Nicolaus Bernoulli introduced the puzzle in 1713. It is named after a leading 18th-century scientific journal, in which Daniel Bernoulli published a paper on it in 1738.
Daniel Bernoulli argued that the value of money grows more slowly than the amount, so huge but unlikely prizes count for less. Strictly speaking, the encyclopedia notes, it is not a paradox in the logical sense, because no formal contradiction is derived. Our page on deciding under uncertainty looks at expected value as a decision tool.
For example, winning a thousand dollars might change your month, while winning a thousand times that would probably not feel a thousand times better.
How sure: Unsettled the encyclopedia sets out several proposed answers, and says the game still produces new puzzles in decision theory.
9. Newcomb’s problem
A reliable predictor has already filled a box based on what you will choose: should you take one box or two?
A transparent box holds one thousand dollars, and an opaque box holds either nothing or one million dollars, depending on a prediction already made about your choice. If the prediction was that you would take both boxes, the opaque box is empty; if it was that you would take just the opaque box, it holds the million.
One rule says take both, because whatever the prediction, two boxes give you one thousand dollars more. Another rule says take one, because the prediction is reliable, so choosing one box goes with a full box. Robert Nozick presented the problem in 1969 and named it after the physicist William Newcomb, who first formulated it.
The puzzle pits two sensible rules against each other: pick what causes the best result, or pick what goes with it. For example, in a game against a near-identical twin, cooperating is a sign, but not a cause, of the twin cooperating too. Our page on game theory explains that kind of game.
How sure: Unsettled decision theorists still defend both rules.
10. The birthday paradox
In a group of just 23 people, the chance that two share a birthday is better than even.
A Cornell maths outreach page puts it this way: in a group of at least 23 randomly chosen people, the chance that some pair shares a birthday is more than 50%. For 57 or more people, it passes 99%.
It feels wrong because we picture someone matching our own birthday. That is a different question from whether any two people in the room match. Our reading: every pair of people is a chance for a match, and each new person can match everyone already there.
The sum assumes all 365 days are equally likely, and real birthdays are uneven, which only makes a match more likely. For example, in a class of 30, a shared birthday is more likely than not.
How sure: Strong it is a standard probability result, under the stated assumption about birthdays.
Puzzles about the night sky
Two puzzles start from a simple look at the sky and end up as questions about the whole universe.
11. The Fermi paradox
If intelligent life should be common in the universe, why have we seen no sign of it?
Enrico Fermi’s question arose over lunch with Emil Konopinski, Edward Teller and Herbert York in the summer of 1950. A 1985 Los Alamos report by E. M. Jones tells the story under the title “Where is everybody?”.
The paradox is the clash between expecting intelligent life elsewhere and the apparently lifeless universe we observe. The expectation is linked to models like the Drake equation: even if life is unlikely at any one place, there are so many places that it should appear somewhere.
A 2018 preprint by Anders Sandberg, Eric Drexler and Toby Ord argues that our uncertainty about the key numbers spans many orders of magnitude. With that uncertainty included, they find a substantial chance that there is no other intelligent life in our observable universe, so silence should be little surprise.
Our reading: a single neat estimate can hide a huge range of possible answers. For example, if you guess every step in a long chain of chances, small errors at each step multiply into a vast range at the end.
How sure: Unsettled the main recent analysis is one team’s 2018 preprint, which has not been published in a journal, so it has not settled the question.
12. Olbers’ paradox: why is the night sky dark?
If the universe went on forever and was filled with stars, every line of sight should end on a star, and the night sky should blaze.
On 7 May 1823, Heinrich Olbers finished a paper concluding that, on what was known then, the night sky should not have been dark. He was not the first to notice the puzzle, but his name is the one attached to it today.
In 1964 the cosmologist Edward Harrison showed that the main factor is the finite age of the stars. A limited number of stars, each burning for a limited time, spread over a huge volume, leaves darkness between them. Harrison later found the same idea in Edgar Allan Poe and in Lord Kelvin, writing in 1901.
The researchers who wrote this history add that the sky is not wholly dark: it faintly glows with the dim relic light of the universe’s past. For example, in an endless forest every line of sight would end at a tree. In a small wood, you can see gaps.
How sure: Strong the authors say observations in the 1980s confirmed the answer.
Where to learn more
- Buridan’s ass, the donkey stuck between two equal piles of hay: our page on mental models from philosophy.
- Simpson’s paradox, when the total says the opposite: our page on numbers that mislead.
- Expected value and other decision tools: our page on deciding under uncertainty.
- The prisoner’s dilemma and how people cooperate: our page on game theory in plain words.
- For depth on any card, the Stanford Encyclopedia of Philosophy entries listed below are free to read and go much further.
Quick answers
What is a paradox in simple terms?
An argument that starts from ideas that seem true, moves by steps that seem fine, and still lands somewhere that seems false.
Has Zeno's paradox been solved?
The usual answer uses mathematics of infinite sums, worked out in the 1800s. The encyclopedia calls it the received view.
What is the Fermi paradox?
The clash between expecting intelligent life elsewhere and the apparently lifeless universe we observe. One 2018 preprint argues the silence is not surprising, but it is one team's analysis.
Is the liar paradox solved?
There is no single accepted answer. Thousands of years have produced a great number of proposed solutions, and the encyclopedia reviews ideas about how the paradox might be resolved.
Use this
Our suggestion: three habits these puzzles can train, for any argument that seems to prove too much.
Which step only looks safe?
Quick quiz
Nothing is scored or saved. Recalling it once now is what makes it last.
1. Why does Zeno say Achilles never catches the tortoise?
Answer: Each time Achilles reaches where the tortoise was, it has moved a little further on.Read it again ↑
2. How many randomly chosen people does it take for a better-than-even chance that two share a birthday?
Answer: 23.Read it again ↑
3. What did Edward Harrison show is the main reason the night sky is dark?
Answer: The stars have a finite age.Read it again ↑
How we checked this · Last checked November 2026
- We read the Stanford Encyclopedia of Philosophy entries on each puzzle, the record of the 1985 Los Alamos report on Fermi's question, the abstract of the 2018 Fermi preprint, and an article on the history of the dark-sky question written by astronomers who research the light of the night sky.
- The Stanford Encyclopedia has no entry on the Fermi paradox, so that card rests on the Los Alamos report and the 2018 preprint, which is one team's analysis.
- Thought experiments such as the Ship of Theseus have their own page. Buridan's ass and Simpson's paradox are covered on our pages on mental models from philosophy and numbers that mislead.