No matter how many articles I read on this subject I cannot comprehend how it proves what it proves. I do well with words and rhetorics, philosophy and science – but as soon as you add numbers my mind goes blank. Not very helpful when those fields often rely on equations and models for explanations and proof. I can somewhat understand equations if explained in a simple or cohesive way – but if at all possible analogies or just word-centric explanations would be very helpful.
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Completeness would be “all true statements have proofs”. Godel constructed a statement which, if true, means that the statement does not have a proof. If the statement is true, it has no proof, thus an incomplete system. If the statement if false, it is a true statement with a proof and thus inconsistent. Any sufficiently expressive logical system is either incomplete or inconsistent. If it can express “This statement has no proof” then it is incomplete.
Mathematicians were bummed. They asked “Can we find all of the statements with proofs?” Alan Turing described a theorem proving machine — a Turing Machine — which would test a statement to see if a proof could be found for it. Turing showed that we can’t tell if the machine would ever stop trying to prove some statements. So we can’t tell if a statement does not have a proof.
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