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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The core statement is something like “no self-consistent recursive axiomatic system can contain a proof of its own validity”. The problem goes something like this:
Q: Math seems great, and I can do all kinds of useful stuff with it, but can I trust it? Is there some way to prove that math is logically valid?
M: Yes – the trouble is, the proof doesn’t use math; it uses this other logical system that I’ll call L1.
Q: Cool. L1 actually seems neat too, and it can do some useful stuff that math can’t. But wait, can I trust L1? Is there some way to prove that L1 is logically valid?
M: Yes. The trouble is, the proof isn’t based on math or L1; it uses this other system called L2.
Q: Can I trust L2?
M: Totally. As long as you accept this proof that’s based on L3…
…and so on like that. All this arose when a mathematician named David Hilbert suggested that we should carefully come up with proofs of all our math ideas, so we know everything’s on firm logical ground, and the community tried for years and went “man, this is harder than we thought it’d be”. And finally Godel did some work and announced “stop trying, I’m pretty sure it’s impossible”.
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