And the same is presumably true if one works with essentially any of what are normally considered standard mathematical functions.

So from this it follows that in at least some instances the axioms of arithmetic can never be used to give a finite proof of whether or not the statement is true.

If two elements are removed at each step, however, then this is no longer true.

With s = 2 and n from 0 to 7 the number of these True for all values of variables is {0, 0, 4, 0, 80, 108, 2592, 7296} , with the first few distinct ones being (see page 781 ) {(p ⊼ p) ⊼ p, (((p ⊼ p) ⊼ p) ⊼ p) ⊼ p, (((p ⊼ p) ⊼ p) ⊼ q) ⊼ q} The number of unequal expressions obtained is {2, 3, 3, 7, 10, 15, 12, 16} (compare page 1096 ), with the first few distinct ones being {p, p ⊼ p, p ⊼ q, (p ⊼ p) ⊼ p, (p ⊼ q) ⊼ p, (p ⊼ p) ⊼ q} Most of the axioms from page 808 are too long to appear early in the list of theorems.

For it is certainly true that cellular automata have many special features.

And indeed if this were true, then it would imply that typical initial conditions would inevitably involve random digit sequences.

And what we will find in this section is that the reverse is also true: continuous models can sometimes yield behavior that appears discrete.

And indeed it is true that in materials that consist of many separate crystals or grains, fractures often tend to follow the boundaries between such elements.

It is no longer true that their evolution leads only to simple transformations of the initial conditions.

And I very strongly suspect that this will also be true of space.