Incompleteness
Gödel's first incompleteness theorem, published in 1931, says that any consistent formal system with rules strong enough to express basic arithmetic contains statements that are true of the numbers but cannot be proved inside the system. The second says that no such system can prove its own consistency. The proof works by coding statements about proofs as statements about numbers, so the system can be made to describe its own workings. The theorems are about formal systems, not about human minds, and adding more axioms does not escape them.
Why it matters
It settled, in the negative, the question of whether all of mathematics could be reduced to one complete set of rules, and it is the most misquoted result in the subject.
Also written: incompleteness theorems, Gödel incompleteness
Where it comes up
Read about Incompleteness on Wikipedia
Sources
- Gödel's incompleteness theorems Wikipedia
- Gödel's Incompleteness Theorems Stanford Encyclopedia of PhilosophyStates precisely what the theorems do and do not claim, which is worth reading before anything popular.
