What’s your preferred response/solution to ~”problems”(?) of events that have probability zero but occur nevertheless
My impression is that people have generally agreed that this paradox is resolved (=formally grounded) by measure theory. I know enough measure theory to know what it is but haven’t gone out of my way to explore the corners of said paradoxes.
But you might be asking me about it in the framework of Yudkowsky’s measure of optimization. Let’s say the states are the real numbers in [0, 1] and the relevant ordering is the same as the one on the real numbers, and we’re using the uniform measure over it. Then, even though the probability of getting any specific real number is zero, the probability mass we use to calculate bit of optimization power is all the probability mass below that number. In that case, all the resulting numbers would imply finite optimization power. … except if we got the result that was exactly the number 0. But in that case, that would actually be infinitely surprising! And so the fact that the measure of optimization returns infinity bits reflects intuition.
It’s (probably) true that our physical reality has only finite precision
I’m also not a physicist but my impression is that physicists generally believe that the world does actually have infinite precision.
I’d also guess that the description length of (a computable version of) the standard model as-is (which includes infinite precision because it uses the real number system) has lower K-complexity than whatever comparable version of physics where you further specify a finite precision.
My impression is that people have generally agreed that this paradox is resolved (=formally grounded) by measure theory. I know enough measure theory to know what it is but haven’t gone out of my way to explore the corners of said paradoxes.
But you might be asking me about it in the framework of Yudkowsky’s measure of optimization. Let’s say the states are the real numbers in [0, 1] and the relevant ordering is the same as the one on the real numbers, and we’re using the uniform measure over it. Then, even though the probability of getting any specific real number is zero, the probability mass we use to calculate bit of optimization power is all the probability mass below that number. In that case, all the resulting numbers would imply finite optimization power. … except if we got the result that was exactly the number 0. But in that case, that would actually be infinitely surprising! And so the fact that the measure of optimization returns infinity bits reflects intuition.
I’m also not a physicist but my impression is that physicists generally believe that the world does actually have infinite precision.
I’d also guess that the description length of (a computable version of) the standard model as-is (which includes infinite precision because it uses the real number system) has lower K-complexity than whatever comparable version of physics where you further specify a finite precision.