On Exactitude in Science

It is known that no classical diagonal argument -and few classical arguments- are sufficiently strong to decide between P = NP and P != NP. The reason is that these arguments tend to "relativize", meaning that they are equally valid in our world, where computation costs time, and in fantasy-worlds where -thanks to the ability of Turing machines to query oracles- it is for free. But in some of these fantasy-worlds, P = NP, while in others, P != NP... An exception is the halting theorem. An oracular Turing machine can decide whether particular Turing machines will halt on particular inputs, but it cannot decide, in general, whether a Turing machine will halt, because it cannot decide if Turing machines equivalent to itself will halt. This creates a hierarchy of machines, each with a more powerful oracle, and a higher order halting problem. In this note, the argument that the inequality of P and NP is a consequence of this extended form of the halting problem is presented in parable form.