As discussed on page 898 , one can set up operator evolution systems similar to symbolic systems (see page 103 ) that have essentially the same relationship to operator systems as sequential substitution systems do to multiway systems.

The fact that nested digit sequences do not correspond to algebraic numbers follows from work by Alfred van der Poorten and others in the early 1980s.

Yet even in such a case one can always define coordinates that nominally allow the surface to be drawn in a finite way—and the Poincaré disk model used in the pictures below is the standard way of doing this.

(Up to 8 nodes, all 8 trivalent networks have this property; up to 10 nodes 25 of 27 do.) • Girth: the length of the shortest cycle in the network.

Such models inevitably handle only computable reals (in the sense defined above), and can never do computations beyond those possible in ordinary discrete systems. … Introducing other standard mathematical functions as primitives does not usually help much, unless one somehow gives the system the capability to solve any equation immediately (see below ).

One reason that all sequences do not grow forever is that even with perfect randomness, there will be fluctuations, and occasionally n will reach a low value that makes it get stuck in a repetitive sequence. … But since there is no fundamental basis for the randomness approximation, it is still conceivable that a particular value of n exists that does not follow its predictions.

It does not repeat with any short period or show any obvious structure for almost all keys.

One way to do this is by using the Gödel number Product[Prime[i]^list 〚 i 〛 , {i, Length[list]}] .

One approach is to use {1, 1} to indicate the boundary of each block, and then within each block to use all possible digit sequences which do not contain {1, 1} , as in the Fibonacci number system discussed on page 892 .

And after mathematics began to show extensive success in describing nature in the late 1600s no further large-scale efforts to do this on the basis of logic appear to have been made.