possible strings of a given length are eventually generated if one starts from the string representing "true".

For if more than half the strings are generated, then somewhere both a string and its negation would have to appear, implying that the system must be inconsistent. And similarly, if less than half the strings are generated, there must be some string for which neither that string nor its negation ever appear, implying that the system is incomplete.

The pictures on the next page show the fractions of strings of given lengths that are generated on successive steps in various multiway systems. In general one might have to wait an arbitrarily large number of steps to find out whether a given string will ever be generated. But in practice after just a few steps one already seems to get a reasonable indication of the overall fraction of strings that will ever be generated.

And what one sees is that there is a broad distribution: from cases in which very few strings can be generated—corresponding to a very incomplete axiom system—to cases in which all or almost all strings can be generated—corresponding to a very inconsistent axiom system.

So where in this distribution do the typical axiom systems of ordinary mathematics lie? Presumably none are inconsistent. And a few—like basic logic and real algebra—are both complete and consistent, so that in effect they lie right in the middle of the distribution. But most are known to be incomplete. And as we discussed above, this is inevitable as soon as universality is present.

But just how incomplete are they? The answer, it seems, is typically not very. For if one looks at axiom systems that are widely used in mathematics they almost all tend to be complete enough to prove at least a fair fraction of statements either true or false.

So why should this be? I suspect that it has to do with the fact that in mathematics one usually wants axiom systems that one can think of as somehow describing definite kinds of objects—about which one then expects to be able to establish all sorts of definite statements.

And certainly if one looks at the history of mathematics most basic axiom systems have been arrived at by starting with objects—such as finite integers or finite sets—then trying to find collections of axioms that somehow capture the relevant properties of these objects.

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