It has been proved (by Kurt Godel) that there are problems in mathematics and logic that are true but cannot be proved (ever). Does this mean mathematics and logic are "broken"?
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hijklmno
ANSWERS: 6
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You're wrong about the 'ever'. It depends on your axioms. Godel showed that any mathematical system containing arithemetic contains statements that cannot be be proved within the system. But all this means in practice is that the "unprovable" statement is one that can be true or false given your initial axioms. it's up to you. Once you decide on the answer, this becomes an axiom of a new system. For instance, Euclid's fifth postulate, that parallel lines never meet, can't be proved from the others. So you can choose it to be true or false. Choose it true, and you get "Euclidean" geometry. Choose it false and you get other geometries, such as "Hyperbolic" and "Spherical". So all Godel means that mathematics and logic are always incomplete and ready to be extended in new and interesting ways. But not broken. Broken would mean inconsistant. Fortuantely axiom systems can be proved to be consistant in that they cannot cause a contradiction.
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No. Logic and mathematics are tools, and not all tools are designed for all tasks. They are still well suited for the the majority of tasks they are used for.
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Maybe, but I'll be damned if we can prove that ;)
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No, but it does mean that there cannot be a system of mathematics that contains all mathematical truths.
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Trying to explain what Godel proved is a bit daunting-I would say that even most mathematicians misunderstand it as they have not analyzed the actual theorem, much less studied the proof. The proof is a toughie & brilliant. The actual result-in my opinion-is rather trivial, & this is a very controversial opinion! He proves that a specific theorem is either true & unprovable from w/i the system, or that the system is inconsistant. And here it is: Theorem: there exists a theorem w/i this system that is true & unprovable(from w/i). The self referential nature of the proof does not make it trivial, but someone needs to show it true for some other theorem to convince me that it is nontrivial. As to your original Q as stated, it is clearly not true. Suppose you present me with Theorem A which you have shown to be unprovable(that is, it cannot be proven true or false).But then there cannot be a counterexample-o/w it would be proven false. Thus, it must be true. So any theorem which you can prove unprovable, I can immediately prove to be true!
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Not necessarily "broken" but they are not the panaceas that your average Andy atheist tries to make them out to be.
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