Extremely Counterfactual Mugging or: the gist of Transparent Newcomb

Bongo

Omega will either award you $1000 or ask you to pay him $100. He will award you $1000 if he predicts you would pay him if he asked. He will ask you to pay him $100 if he predicts you wouldn't pay him if he asked.

Omega asks you to pay him $100. Do you pay?

This problem is roughly isomorphic to the branch of Transparent Newcomb (version 1, version 2) where box B is empty, but it's simpler.

Here's a diagram:

Omega will either award you $1000 or ask you to pay him $100. He will award you $1000 if he predicts you would pay him if he asked. He will ask you to pay him $100 if he predicts you wouldn't pay him if he asked.

Omega asks you to pay him $100. Do you pay?

This problem is roughly isomorphic to the branch of Transparent Newcomb (version 1, version 2) where box B is empty, but it's simpler.

Here's a diagram:

You've added an extra axiom to Omega, noted that this resulted in a consistent result, and concluded that therefore the original axioms are incomplete (because the result is changed).

But that does not follow. This would only be true if the axiom was added secretly, and the result was still consistent. But because I know about this extra axiom, you've changed the problem; I behave differently, so the whole setup is different.

Or consider a variant: I have the numbers sqrt[2], e and pi. I am required to output the first number that I can prove is irrational, using the shortest proof I can find. This will be sqrt[2] (or maybe e), but not pi. Now add the axiom "pi is irrational". Now I will output pi first, as the proof is one line long. This does not mean that the original axiomatic system was incorrect or under-specified...

I'm not completely sure what your comment means. The result hasn't "changed", it has appeared. Without the extra axiom there's not enough axioms to nail down a single result (and even with it I had to resort to lexicographic chance at one point). That's what incompleteness means here.

If you think that's wrong, try to prove the "correct" result, e.g. that any agent who precommits to not paying won't get the $1000, using only the original axioms and nothing else. Once you write out the proof, we will know for certain that one of us is wrong or the original axioms are inconsistent, which would be even better :-)