In the early 1900s it was widely believed that this would effectively be the case in all reasonable mathematical axiom systems. For at the time there seemed to be no limit to the power of mathematics, and no end to the theorems that could be proved.

But this all changed in 1931 when Gödel's Theorem showed that at least in any finitely-specified axiom system containing standard arithmetic there must inevitably be statements that cannot be proved either true or false using the rules of the axiom system.

This was a great shock to existing thinking about the foundations of mathematics. And indeed to this day Gödel's Theorem has continued to be widely regarded as a surprising and rather mysterious result.

But the discoveries in this book finally begin to make it seem inevitable and actually almost obvious. For it turns out that at some level it can be viewed as just yet another consequence of the very general Principle of Computational Equivalence.

So what is the analog of Gödel's Theorem for multiway systems? Given the setup on page 780 one can ask whether a particular multiway system is complete in the sense that for every possible string the system eventually generates either that string or its negation.

And one can see that in fact the third multiway system is incomplete, since by following its rules one can never for example generate either or its negation . But what if one extends the rules by adding more transformations, corresponding to more axioms? Can one always in the end make the system complete?

If one is not quite careful, one will generate too many strings, and inevitably get inconsistencies where both a string and its negation appear, as in the second picture on the facing page. But at least if one only has to worry about a limited number of steps, it is always possible to set things up so as to get a system that is both complete and consistent, as in the third picture on the facing page.

And in fact in the particular case shown on the facing page it is fairly straightforward to find rules that make the system always complete and consistent. But knowing how to do this requires having behavior that is in a sense simple enough that one can foresee every aspect of it.

From Stephen Wolfram: A New Kind of Science [citation]