Learning path
What Is Truth?
A single sentence about itself brings down a 2,000-year-old assumption, then takes set theory and finally arithmetic with it. This is the shortest route from a schoolyard riddle to the limits of mathematics.
The Liar Paradox
Four words — "this sentence is false" — and there is still no agreed way out. Everything downstream is a response to this.
"This sentence is false." If it is true then it is false, and if it is false then it is true. Two and a half thousand years later there is still no agreed answer — only competing bills to pay, in logic, in language, or in common sense.
AdvancedLogical contradiction26 min
The Barber Paradox
A deliberate detour, and the most useful one on this path. The barber looks exactly like the Liar and dissolves in a second. Learning why it is easy is how you understand why the Liar is not.
A barber shaves everyone who does not shave himself, and nobody else. So who shaves the barber? The honest answer is not a paradox at all — it is a proof that no such barber can exist, and that difference is the whole lesson.
BeginnerApparent contradiction11 min
Berry's Paradox
The same trouble moved from truth to definability. Two different words break in the same way, which suggests the problem is self-reference itself.
"The smallest positive integer not definable in under sixty letters." That phrase has fifty-seven letters, so it defines the very number it says cannot be defined. The way out is that "definable" cannot mean what it seems to.
AdvancedLogical contradiction19 min
Russell's Paradox
Now it stops being a curiosity. The same move destroys the foundation Frege had just spent decades building, and the letter announcing it is one of the most consequential in the history of mathematics.
Collect every set that does not contain itself. Does that collection contain itself? Either answer contradicts the other. The question destroyed the leading attempt to found mathematics on logic, and forced set theory to be rebuilt.
AdvancedLogical contradiction19 min
Gödel's Incompleteness Theorems
The summit. Gödel takes the Liar sentence, swaps "true" for "provable", and turns a paradox into a theorem: any system strong enough to describe arithmetic contains truths it cannot prove.
Any consistent set of rules rich enough to describe arithmetic contains true statements it cannot prove — and cannot prove its own consistency. A precise theorem about formal systems, and the most misquoted result in mathematics.
ExpertImpossibility result25 min
The Halting Problem
The same argument, one more time, in machines instead of sentences. Once you have seen it here you will recognise the diagonal move anywhere.
No program can look at any other program and always correctly say whether it will finish running or loop forever. Turing proved it in 1936 by building a program that does the opposite of whatever it is told about itself.
AdvancedImpossibility result20 min
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