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Fiction "Division by Zero" By Ted Chiang

I read the fiction story "Division by Zero" By Ted Chiang

My interpretation is the character finds a proof that arithmetic is inconsistent.

Is there a formal proof the fiction can't come true? (I don't suggest the fiction can come true).

EDIT: I see someone tried

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+1 for interesting question. I'll have to finish reading it first. But I do find a bit to quibble with in the first paragraph: usually $0\times \infty$ is regarded as an indeterminate form, not 0. –  Willie Wong Dec 18 '10 at 14:15
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This might be interesting to you... –  J. M. Dec 18 '10 at 14:17
    
Ted Chiang resources: freesfonline.de/authors/Ted_Chiang.html –  jerr18 Dec 18 '10 at 14:33
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As with all stories, it's about people, emotions etc. Any math it borrows is just a prop, and doesn't have to be correct. As with many stories, there is a kind of superficial profoundness. You may as well ask if there's a proof that Gandalf can't come back from the dead as Gandalf the Green, Gandalf the Tartan etc - there is no formal proof of any such thing, but it's still not going to happen in reality. –  Steve314 Dec 18 '10 at 16:46
    
Possibly worth saying, though - there are some people who claim it's possible to taking a ratio of the largest useful number to the smallest useful number based on physics and astronomy, and give infinity a finite value. Do this and arithmetic is trivially proven to be self-contradictary. –  Steve314 Dec 18 '10 at 17:01

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up vote 7 down vote accepted
Is there a formal proof the fiction can't come true?

No, by Gödel's second incompleteness theorem, formal systems can prove their own consistency if and only if they are inconsistent. So given that arithmetic is consistent, we'll never be able to prove that it is. (EDIT: Actually not quite true; see Alon's clarification below.)

As an aside, if you liked "Division by Zero," you might also like Greg Egan's pair of stories in which arithmetic isn't consistent: "Luminous" and "Dark Integers".

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Given that arithmetic is consistent, then it (arithmetic) cannot prove that it is. That doesn't mean that some other system can't achieve that feat. For instance, the consistency of Peano Arithmetic, which is a quite reasonable form of "Arithmetic", can be proved in ZFC, and much weaker systems as well. –  Alon Amit Dec 19 '10 at 10:18
    
Is this scenario possible: No matter if arithmetic is consistent or not, it can prove that a counterexample of its consistency can't be constructed, well, within itself (sorry if that doesn't make any sence, I mean, a counterexample will need more axioms). –  jerr18 Dec 20 '10 at 18:19

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