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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Suppose you have some formal, logical theory of arithmetic. That is, you’ve reduced statements like 2 + 2 = 4 (a true statement) or 5 – 4 = 7 (a false one) into purely logical terms, without directly invoking the concept of numbers.
The (first) incompleteness theorem says that, given such a language, there are sensible statements you can make, but which can’t be proven true *or* proven false using that logical language. In other words, you can never create a structured theory of arithmetic (satisfying certain relatively weak assumptions) that can prove every true statement in arithmetic.
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