The Barber Paradox
The barber shaves every man in town who does not shave himself, and no one else. So who shaves the barber?
1918 (Russell's lectures); the story circulated earlier11 min read
Domain
Logic
Self-reference, contradiction, and the limits of formal systems.
16 entriesFilter these further →
Beginner
Ages 8+, no background needed.
Intermediate
Curious teenagers and general adult readers.
The Grandfather Paradox
Travel back in time and stop your own grandparents from ever meeting. Then who was it that went back?
1943 (Barjavel's novel), with antecedents from the 1920s18 min read
The Paradox of the Court
A student agrees to pay for his lessons the day he wins his first case. He never takes a case. His teacher sues him for the money.
c. 420 BCE (traditional); earliest surviving account 2nd century CE20 min read
The Two Generals' Problem
Two armies must attack at the same moment. Any message can be lost, including the one confirming the last one arrived.
1975 (proved), 1978 (named)18 min read
The Grelling-Nelson Paradox
Is the word "heterological" heterological? Either answer generates a contradiction.
19087 min read
Advanced
Undergraduate level in the relevant field.
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
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
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 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 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
Expert
Graduate level or specialist readers.
Gödel's Incompleteness Theorems
In any system of arithmetic strong enough to be useful, there are true statements it can never prove. Including this one.
193125 min read
The FLP Impossibility Result
A handful of computers cannot be guaranteed to agree on a single value if even one of them might be slow. Not merely hard — impossible.
198523 min read