“There is another point that should be made, however, and this is that it need not be the case that human mathematical understanding is in principle as powerful as any oracle machine at all. As noted above, the conclusion G does not necessarily imply that human insight is powerful enough, in principle, to solve each instance of the halting problem. Thus, we need not necessarily conclude that the physical laws that we seek reach, in principle, beyond every computable level of of oracle machine (or even reach the first order). We need only seek something that is not equivalent to any specific oracle machine (including also the zeroth-order machines, which are Turing machines). Physical laws could perhaps lead to something that is just different.”