Over the past seventy years a few simpler examples have been constructed—mostly with no obviously self-referential character.
But usually these examples have involved rather sophisticated and obscure mathematical constructs—most often functions that are somehow set up to grow extremely rapidly. Yet at least in principle there should be examples that can be constructed based just on statements that no solutions exist to particular integer equations.
If an integer equation such as x2y3+12 has a definite solution such as x47, y13 in terms of particular finite integers then this fact can certainly be proved using the axioms of arithmetic. For it takes only a finite calculation to check the solution, and this very calculation can always in effect be thought of as a proof.
But what if the equation has no solutions? To test this explicitly one would have to look at an infinite number of possible integers. But the point is that even so, there can still potentially be a finite mathematical proof that none of these integers will work.
And sometimes the proof may be straightforward—say being based on showing that one side of the equation is always odd while the other is always even. In other cases the proof may be more difficult—say being based on establishing some large maximum size for a solution, then checking all integers up to that size.
And the point is that in general there may in fact be absolutely no proof that can be given in terms of the normal axioms of arithmetic.
So how can one see this?
The picture on the facing page shows that one can construct an integer equation whose solutions represent the behavior of a system like a cellular automaton. And the way this works is that for example one variable in the equation gives the number of steps of evolution, while another gives the outcome after that number of steps.
So with this setup, one can specify the number of steps, then solve for the outcome after that number of steps. But what if for example one instead specifies an outcome, then tries to find a solution for the number of steps at which this outcome occurs?
If in general one was able to tell whether such a solution exists then it would mean that one could always answer the question of