Let me tell you a theorem whose proof is so long you couldn't fit it in the observable Universe if you wrote one symbol on every electron, proton and neutron available. In fact.... (to be continued)pic.twitter.com/Uq93Exq27f
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Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.
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Let me tell you a theorem whose proof is so long you couldn't fit it in the observable Universe if you wrote one symbol on every electron, proton and neutron available. In fact.... (to be continued)pic.twitter.com/Uq93Exq27f
So, by stepping out of an axiom system, assuming it's consistent and using that to draw conclusions, we can shorten proofs of some results by an arbitrary amount! For some more about the world's longest proofs, try this:https://johncarlosbaez.wordpress.com/2012/10/19/insanely-long-proofs/ …
Of course two questions arise here, one (far) more straightforward than the other. First, are there naturally-occurring statements whose shortest proofs are extremely long in what we all now think are reasonable axiom systems; and, second, what is reasonable, and why?
We could now add the Kepler conjecture to our favorite axioms for math as an extra axiom, and considerably shorten Hales' proof. But this feels like "cheating" - and proving the new axiom system is equivalent to the original one would take a long time.
Yes. So the real (extremely subtle) question is, what axiom should we adopt, to ensure the existence of a proof of reasonable length, without "cheating"? If the shortest proof now really is so long, then we need new axioms. But we want our proofs to teach us something. Tricky.
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