been arrived at by the single rather specific approach of starting from some known set of theorems, then trying to find systems that are progressively more general, yet still manage to satisfy these theorems.
And given this approach, it tends to be the case that the questions that are considered interesting are ones that revolve around whatever theorems a system was set up to satisfy—making it rather likely that these questions can themselves be addressed by similar theorems, without any confrontation with undecidability or unprovability.
But what if one looks at other kinds of systems?
One of the main things I have done in this book is in a sense to introduce a new approach to generalization in which one considers systems that have simple but completely arbitrary rules—and that are not set up with any constraint about what theorems they should satisfy.
But if one has such a system, how does one decide what questions are interesting to ask about it? Without the guidance of known theorems, the obvious thing to do is just to look explicitly at how the system behaves—perhaps by making some kind of picture.
And if one does this, then what I have found is that one is usually immediately led to ask questions that run into phenomena like undecidability. Indeed, from my experiments it seems that almost as soon as one leaves behind the constraints of mathematical tradition undecidability and unprovability become rather common.
As the picture on the next page indicates, it is quite straightforward to set up an axiom system that deals with logical statements about a system like a cellular automaton. And within such an axiom system one can ask questions such as whether the cellular automaton will ever behave in a particular way after any number of steps.
But as we saw in the previous section, such questions are in general undecidable. And what this means is that there will inevitably be cases of them for which no proof of a particular answer can ever be given within whatever axiom system one is using.
So from this one might conclude that as soon as one looks at cellular automata or other kinds of systems beyond those normally studied in mathematics it must immediately become effectively impossible to make progress using traditional mathematical methods.