12.   Numbers in the RUCA

The unit of length, L, in the RUCA is the dis­tance from one cell to its nearest neighbor. Values of L are always numbers that have finite represen­tations. All effective values (or numbers) within a DM system are capable of having a finite repre­sentation. They may be real or complex. This is a simple consequence of the fact that the RUCA is finite; there is not enough information in a finite RUCA to represent even one arbitrary real num­ber (or even a rational number with too large a numerator or denominator). Many real numbers have a finite description, such as "pi" or 2^1/2 Such descriptions of real numbers can exist in a DM system as a consequence of a process which can be shown to asymptotically approach the rep­resentation of the real number. In a DM, the repre­sentation of a number might reach an exact limit in a finite number of steps if the limit has a fi­nite representation, or it might continue to do the digital equivalent of an asymptotic approach to a value with an infinite representation, never reach­ing the limit.

The contemporary way that Achilles catches the tortoise is to use calculus, which involves an infi­nite number of time steps in a finite amount of time. This is not possible in DM. Instead, in DM Achilles chases the tortoise on a chess board where Achilles gets to move to the next square every 3 ticks of the clock, while the tortoise only gets to move every 7 ticks of the clock. In DM, as in the real world, Achilles catches the tortoise. We see the flaw in the calculus solution to Zeno's para­dox as ignoring the possibility that the informa­tional work to calculate whether to take a step (or which way to take a step) is not proportional to the distance to the tortoise. This means that it is not clear how the computational task in making an infinite number of steps can be done in finite time.