You know that automated proof verifiers exist, right? And also that programs can know their own source code? Well, here's a puzzle for you:
Consider a program A that knows its own source code. The algorithm of A is as follows: generate and check all possible proofs up to some huge size (3^^^^3). If A finds a proof that A returns 1, it returns 1. If the search runs to the end and fails, it returns 0. What will this program actually return?
Wait, that was the easy version. Here's a harder puzzle:
Consider programs A and B that both know their own, and each other's, source code. The algorithm of A is as follows: generate and check all proofs up to size 3^^^^3. If A finds a proof that A returns the same value as B, it returns 1. If the search fails, it returns 0. The algorithm of B is similar, but possibly with a different proof system and different length limit. What will A and B return?
This second puzzle is a formalization of a Prisoner's Dilemma strategy proposed by Eliezer: "I cooperate if and only I expect you to cooperate if and only if I cooperate". So far we only knew how to make this strategy work by "reasoning from symmetry", also known as quining. But programs A and B can be very different - a human-created AI versus an extraterrestrial crystalloid AI. Will they cooperate?
I may have a tentative proof that the answer to the first problem is 1, and that in the second problem they will cooperate. But: a) it requires you to understand some logic (the diagonal lemma and Löb's Theorem), b) I'm not sure it's correct because I've only studied the subject for the last four days, c) this margin is too small to contain it. So I put it up here. I submit this post with the hope that, even though the proof is probably wrong or incomplete, the ideas may still be useful to the community, and someone else will discover the correct answers.
Edit: by request from Vladimir Nesov, I reposted the proofs to our wiki under my user page. Many thanks to all those who took the time to parse and check them.
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.