Gödel's Incompleteness (extra footage 1) - Numberphile
MAIN VIDEO: https://youtu.be/O4ndIDcDSGc More links & stuff in full description below ↓↓↓ Extra footage part 2: https://youtu.be/7DtzChPqUAw Professor ...
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The theorem implies that there will be an infinite number of undecidable sentences, sentences which are true which cannot be proved. So, because then, you might hope, well, we could just add those finite number as axioms and then we will have a complete. This is, the incompleteness is referring to the fact that there's this kind of gap between truth and proof. We'd love a complete system where we've got a set of axioms, and all truths can be proved. So incompleteness refers to the fact that you can never kind of complete it. Brady: "How will a mathematician ever know if what they're working on is truly undecidable, or falls into the Gödel basket? "It seems like you could have things in your too hard basket, and you would say, ah, this must be a Gödel problem.&qu...
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The theorem implies that there will be an infinite number of undecidable sentences, sentences which are true which cannot be proved. So, because then, you might hope, well, we could just add those finite number as axioms and then we will have a complete. This is, the incompleteness is referring to the fact that there's this kind of gap between truth and proof. We'd love a complete system where we've got a set of axioms, and all truths can be proved. So incompleteness refers to the fact that you can never kind of complete it. Brady: "How will a mathematician ever know if what they're working on is truly undecidable, or falls into the Gödel basket? "It seems like you could have things in your too hard basket, and you would say, ah, this must be a Gödel problem.&qu...
