Maxwell's Demon
A tiny creature sorting fast molecules from slow ones appears to make heat flow the wrong way, for free, forever.
1867 (letter to Tait), 1871 (published in Theory of Heat)18 min read
Difficulty
Advanced
Assumes some formal or technical background at depth.
Written for: Undergraduate level in the relevant field.
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Russell's Paradox
Consider the set of all sets that do not contain themselves. Does that set contain itself? Either answer is wrong.
1901 (discovered), 1902 (letter to Frege), 1903 (published)19 min read
The Halting Problem
No program can ever be written that reliably tells you whether another program will finish running. This is not a limit of our cleverness.
193620 min read
The Liar Paradox
"This sentence is false." Decide whether you are looking at something true or something false, and see how far you get.
c. 600 BCE (the Epimenides line); c. 350 BCE (Eubulides' paradox)26 min read
Zeno's Dichotomy Paradox
To cross a room you must first cross half of it, then half of what is left, and so on without end. So how does anyone move?
c. 450 BCE18 min read
Braess's Paradox
Add a new road to a congested city and every journey gets slower. Close the road again and the traffic speeds up.
196819 min read
Cantor's Diagonal Argument
There are more numbers between 0 and 1 than there are whole numbers in all of eternity. Both lists are infinite.
189116 min read
Newcomb's Problem
The boxes are already filled. Does choosing one still change what is in them?
c. 1960 (posed), 1969 (published)10 min read
The Allais Paradox
Offered the same improvement in odds twice, most people take it once and refuse it the second time. Both choices feel obviously right.
1952 (posed at the Paris colloquium), 1953 (published)22 min read
The Byzantine Generals Problem
Several generals must attack together or be destroyed. They can only send messengers, and some of the generals are traitors.
198222 min read
The CAP Theorem
A system spread across several machines can give you consistent answers, or always answer at all. The network makes you choose.
2000 (conjectured), 2002 (proved)23 min read
The Sleeping Beauty Problem
Woken with no memory of having been woken before, how confident should you be that the coin came up heads?
1997 (posed), 2000 (named and popularised)24 min read
The Sorites Paradox
One grain of sand is not a heap. Adding a single grain never turns a non-heap into a heap. So heaps do not exist.
c. 350 BCE20 min read
The St. Petersburg Paradox
A coin-flipping game whose average winnings are infinite. How much of your own money would you actually pay to play?
1713 (posed by Nicolaus Bernoulli), 1738 (published by Daniel Bernoulli)19 min read
The Two Envelopes Problem
One envelope holds twice as much money as the other. The arithmetic says swap — then says swap back, forever.
1953 (Littlewood, crediting Schrödinger), 1989 (envelope form)19 min read
The Unexpected Hanging Paradox
A prisoner is told he will hang next week, on a day he cannot possibly predict. He proves it cannot happen. It happens.
1943 (in circulation), 1948 (first in print)21 min read
Berry's Paradox
"The smallest number that cannot be described in fewer than twelve words." That description is eleven words long.
1906 (published by Russell, crediting Berry); communicated c. 190419 min read
Levinthal's Paradox
A protein has more possible shapes than the universe has atoms, yet it finds the right one in less than a thousandth of a second.
1968–196917 min read
The Giffen Paradox
Some things sell more as the price goes up, and the people buying them are behaving perfectly sensibly.
1895 (Marshall's third edition); 2008 (first credible field evidence)18 min read
The Ladder Paradox
A ladder too long to fit inside a barn fits inside it — and also does not — depending entirely on who is watching.
1905 (the physics), 1961 (first prominent published version)19 min read
The Productivity Paradox
Computers arrived in every office through the 1970s and 80s. Measured productivity did not move for two decades.
1987 (Solow's remark), debate ongoing17 min read
The Ross-Littlewood Paradox
You add ten balls and remove one, infinitely many times. At the end, the vase is empty. What happened to all the balls?
19887 min read