For the purposes of this post, we are not much concerned about the constraints of our specific reality, for 3 reasons:
If we accept this argument, then it’s unlikely that a true Universal Turing Machine exists in real life either, because it doesn’t have time bounds or memory bounds. So we are already idealizing things away to make it work at all.
If we did use real humans computing/counting things, rather than idealized human computing/counting things, then the problem becomes the fact that we only have a finite, constant memory and time, and the functions/algorithms that are in the computable set as defined by a UTM does not collapse to the set of functions calculable by a human. If they did, such that human people in real life could calculate all the computable functions in constant time and memory, that would violate the time and space hierarchy theorems. I literally addressed this.
While we think that halting oracles can’t exist, and there are many good reasons to believe that, it doesn’t totally mean that we figured things out yet, and while I basically agree that it’s unlikely, I am unwilling to make very firm conclusions about the very far future, including this one.
For the purposes of this post, we are not much concerned about the constraints of our specific reality, for 3 reasons:
If we accept this argument, then it’s unlikely that a true Universal Turing Machine exists in real life either, because it doesn’t have time bounds or memory bounds. So we are already idealizing things away to make it work at all.
If we did use real humans computing/counting things, rather than idealized human computing/counting things, then the problem becomes the fact that we only have a finite, constant memory and time, and the functions/algorithms that are in the computable set as defined by a UTM does not collapse to the set of functions calculable by a human. If they did, such that human people in real life could calculate all the computable functions in constant time and memory, that would violate the time and space hierarchy theorems. I literally addressed this.
While we think that halting oracles can’t exist, and there are many good reasons to believe that, it doesn’t totally mean that we figured things out yet, and while I basically agree that it’s unlikely, I am unwilling to make very firm conclusions about the very far future, including this one.