If it's purely theoretical then why can't I have a hypercomputer? What's wrong with simply solving the halting problem by using an oracle, or by running a Turing machine for infinitely many steps before I make my decision?
If I can't have infinite time, then I might as well have 5 seconds.
If it's purely theoretical then why can't I have a hypercomputer? What's wrong with simply solving the halting problem by using an oracle, or by running a Turing machine for infinitely many steps before I make my decision?
You're asking the same question three times.
Anyway, an oracle can determine whether a program in a Turing machine can halt. It can't determine whether it itself can halt.
Any attempt to use an oracle could lead to X predicting Y who tries to predict X using an oracle. That can be equivalent to the oracle trying to determine whether it...
Person X stands in front of a sophisticated computer playing the decision game Y which allows for the following options: either press the button "sim" or "not sim". If she presses "sim", the computer will simulate X*_1, X*_2, ..., X*_1000 which are a thousand identical copies of X. All of them will face the game Y* which - from the standpoint of each X* - is indistinguishable from Y. But the simulated computers in the games Y* don't run simulations. Additionally, we know that if X presses "sim" she receives a utility of 1, but "not sim" would only lead to 0.9. If X*_i (for i=1,2,3..1000) presses "sim" she receives 0.2, with "not sim" 0.1. For each agent it is true that she does not gain anything from the utility of another agent despite the fact she and the other agents are identical! Since all the agents are identical egoists facing the apparently same situation, all of them will take the same action.
Now the game starts. We face a computer and know all the above. We don't know whether we are X or any of the X*'s, should we now press "sim" or "not sim"?
EDIT: It seems to me that "identical" agents with "independent" utility functions were a clumsy set up for the above question, especially since one can interpret it as a contradiction. Hence, it might be better to switch to identical egoists whereas each agent only cares about her receiving money (linear monetary value function). If X presses "sim" she will be given 10$ (else 9$) in the end of the game; each X* who presses "sim" receives 2$ (else 1$), respectively. Each agent in the game wants to maximize the expected monetary value they themselves will hold in their own hand after the game. So, intrinsically, they don't care how much money the other copies make.
To spice things up: What if the simulation will only happen a year later? Are we then able to "choose" which year it is?