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Quantum Computing since Democritus by Scott Aaronson

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3 Gödel, Turing, and friends

In the last chapter, we talked about the rules for first-order logic. There's an amazing result called Gödel's Completeness Theorem that says that these rules are all you ever need. In other words: if, starting from some set of axioms, you can't derive a contradiction using these rules, then the axioms must have a model (i.e., they must be consistent). Conversely, if the axioms are inconsistent, then the inconsistency can be proved using these rules alone.

Think about what that means. It means that Fermat's Last Theorem, the Poincaré Conjecture, or any other mathematical achievement you care to name can be proved by starting from the axioms for set theory, and then applying these piddling little rules over and over ...

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