But while one can emulate each step in the evolution of a mobile automaton or a Turing machine with a single step of cellular automaton evolution, this is no longer in general true for substitution systems.

Non-standard arithmetic Goodstein's result from page 1163 is true for all ordinary integers.

It is true that in studying questions related to continuous mathematics, imprecise numerical approximations have often been made when computers are used (see above ).

The same is true for higher-dimensional generalizations such as so-called Anosov maps {x, y}  Mod[m .

For there one tends to think not so much about transforming expressions as about taking collections of true statements (such as equations u  v ), and using so-called rules of inference to deduce other ones. … And with this approach axioms enter merely as initial true statements, leaving rules of inference to generate successive steps in proofs.

But when the wave theory of light finally became popular in the mid-1800s it seemed to imply that no similar principle could be true for light. … Then in 1905 Albert Einstein proposed his so-called special theory of relativity—which took as its basic postulates not only that the laws of mechanics and electrodynamics are independent of how fast one is moving, but that this is also true of the speed of light.

But this is not true of every number.

What is true, however, is that the phenomenon was immensely easier to discover in discrete systems than it would have been in continuous ones.

It turns out that the same is true for nested sequences.

And what this means is that almost any statement that can, for example, readily be investigated by the traditional methods of mathematical proof will tend to be largely irrelevant to the true Principle of Computational Equivalence.