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orthonormal comments on AI cooperation in practice - Less Wrong
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Comments (157)
I believe Program A in "the easy version" would return 0. Assuming zero run-time errors, its structure implements the logical structure:
However n is defined (the post proposes n = 3^^^^3), it can be shown by the definition of the word "proof" that:
and therefore the first proposition holds for every program, and cannot be used to show that A returns 1.
However, the second proposition also cannot be used to show that A returns 1. If the given condition holds, A does not return 1; if the given condition does not hold, the second proposition demonstrates nothing.
Therefore no property of the program can be used to demonstrate that the program must return 1. Therefore no proof can demonstrate that the program must return 1. Therefore the program will find no proof that the program returns 1. Therefore the program will return 0.
Q.E.D.
IIRC, the modification of Gödel's statement which instead has the interpretation "I can be proved in this formal system" is called a Henkin sentence, and does in fact have a finite proof in that system. This seems weird in the intuitive sense you're talking about, but it's actually the case.
Yep. The Henkin sentence has already come up multiple times in the comments here. Understanding the proof of the Henkin sentence takes you about 95% of the way to understanding my original argument, I think.