Someone may already have mentioned this, but doesn’t the fact that these scenarios include self-referencing components bring Goedel’s Incompleteness Theorem into play somehow? I.e. As soon as we let decision theories become self-referencing, it is impossible for a “best” decision theory to exist at all.
One important thing to consider is that there may be a sensible way to define “best” that is not susceptible to this type of problem. Most notably, there may be a suitable, solvable, and realistic subclass of problems over which to evaluate performance. Also, even if there is no “best”, there can still be better and worse.
doesn’t the fact that these scenarios include self-referencing components bring Goedel’s Incompleteness Theorem into play somehow?
Self-reference and the like is necessary for Goedel sentences but not sufficient. It’s certainly plausible that this scenario could have a Goedel sentence, but whether the current problem is isomorphic to a Goedel sentence is not obvious, and seems unlikely.
Perhaps referring directly to Goedel was not apt. What Goedel showed was that Hilbert/Russell’s efforts were futile. And what Hilbert and Russell were trying to do was create a formal system where actual self-reference was impossible. And the reason he was trying to do that, finally, was that self-reference creates paradoxes which reduce to either incompleteness or inconsistency. And the same is true of these more advanced decision theories. Because they are self-referencing, they create an infinite regress that precludes the existence of a “best” decision theory at all.
So, finding a best decision theory is impossible once self-reference is allowed, because of the nature of self-reference, but not quite because of Goedel’s theorems, which are the stronger declaration that any formal system by necessity contains self-referential aspects that make it incomplete or inconsistent.
Someone may already have mentioned this, but doesn’t the fact that these scenarios include self-referencing components bring Goedel’s Incompleteness Theorem into play somehow? I.e. As soon as we let decision theories become self-referencing, it is impossible for a “best” decision theory to exist at all.
There was some discussion of much the same point in this comment thread
One important thing to consider is that there may be a sensible way to define “best” that is not susceptible to this type of problem. Most notably, there may be a suitable, solvable, and realistic subclass of problems over which to evaluate performance. Also, even if there is no “best”, there can still be better and worse.
Self-reference and the like is necessary for Goedel sentences but not sufficient. It’s certainly plausible that this scenario could have a Goedel sentence, but whether the current problem is isomorphic to a Goedel sentence is not obvious, and seems unlikely.
Perhaps referring directly to Goedel was not apt. What Goedel showed was that Hilbert/Russell’s efforts were futile. And what Hilbert and Russell were trying to do was create a formal system where actual self-reference was impossible. And the reason he was trying to do that, finally, was that self-reference creates paradoxes which reduce to either incompleteness or inconsistency. And the same is true of these more advanced decision theories. Because they are self-referencing, they create an infinite regress that precludes the existence of a “best” decision theory at all.
So, finding a best decision theory is impossible once self-reference is allowed, because of the nature of self-reference, but not quite because of Goedel’s theorems, which are the stronger declaration that any formal system by necessity contains self-referential aspects that make it incomplete or inconsistent.