Gödel's Incompleteness Theorems
Kurt Gödel · 1931
"Any formal system powerful enough to describe arithmetic contains true statements it can never prove, and it can't even prove its own consistency."
Gödel was 25 when he proved something unsettling about mathematics: any system rich enough to do arithmetic will always contain true statements it can't prove from within.
Gödel's first theorem shows any consistent formal system capable of arithmetic contains true statements it can neither prove nor disprove — completeness and consistency can't both be had. His second theorem goes further: such a system can't prove its own consistency without stepping outside itself, by encoding 'this statement is unprovable' directly into arithmetic.
What does Gödel's second incompleteness theorem say a consistent formal system capable of arithmetic can never do?
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The explanation above is written with AI assistance. These are the originals — go to them to check it.
- Gödel's Incompleteness TheoremsStanford Encyclopedia of Philosophy
Bayes' Theorem
"Belief should update in proportion to evidence — Bayes' theorem is the exact math for how much a new piece of evidence should move your confidence."