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 :-)
I was also previously suspicious to the word “change”, but lately made my peace with it. Saying that there’s change is just a way of comparing objects of the same category. So if you look at an apple and a grape, what changes from apple to grape is, for example, color. A change is simultaneously what’s different, and a method of producing one from the other. Application of change to time, or to the process of decision-making, are mere special cases. Particular ways of parsing change in descriptions of decision problems can be incorrect because of explicit dependence bias: those changes as methods of determining one from the other are not ambient dependencies. But other usages of “change” still apply. For example, your decision to take one box in Newcomb’s instead of two changes the content of the box.
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 :-)
I was also previously suspicious to the word “change”, but lately made my peace with it. Saying that there’s change is just a way of comparing objects of the same category. So if you look at an apple and a grape, what changes from apple to grape is, for example, color. A change is simultaneously what’s different, and a method of producing one from the other. Application of change to time, or to the process of decision-making, are mere special cases. Particular ways of parsing change in descriptions of decision problems can be incorrect because of explicit dependence bias: those changes as methods of determining one from the other are not ambient dependencies. But other usages of “change” still apply. For example, your decision to take one box in Newcomb’s instead of two changes the content of the box.