If less than half the strings of a given length are ever produced, this means that there must be some strings where neither the string nor its negation can be proved, indicating incompleteness.

exactly the same phenomenon of successively sampling less and less significant digits still occurs.

So even though the systems themselves generate their behavior by going through a whole sequence of steps, we can readily shortcut this process and find the outcome with much less effort. … But to have any kind of traditional theory one must find a shortcut that involves much less computation.

And as technology advances less and less regularity needs to be present.

If this total is less than 4, then the cell becomes white on the next step, while if the total is greater than 6, it becomes black.

For all but code 1599, the fate of these patterns in fact becomes clear after less than 100 steps.

More or less throughout, there is a very regular background texture that consists of an array of small white triangles repeating every 7 steps. … But on the right-hand side, the pattern is much less regular.

As I will discuss in the next chapter , my general expectation is that more or less any system whose behavior is not somehow fundamentally repetitive or nested will in the end turn out to be universal. … And indeed in any system with class 4 behavior, things will typically always work in more or less the same way.

And what this suggests is that it makes no more or less sense to talk about the meaning of phenomena in our universe as it does to talk about the meaning of phenomena in the digit sequence of π .

The number of turns or whorls varies widely, from less than one in a typical bivalve, to more than thirty in a highly pointed univalve such as a screw shell. Usually the coiled structure is obvious from looking at the apex on the outside of the shell, but in cowries, for example, it is made less obvious by the fact that later whorls completely cover earlier ones, and at the opening of the shell some dissolving and resculpting of material occurs.