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But if one works, say, with Implies then there are for example representations of size 3 (see above ). … Indeed, in general with operators Implies , And and Or one gets to 2 n - 1 elements, while with operators Xor and Equal one gets to 2^(2Floor[n/2]) elements.
This particular cellular automaton rule has the special property of additivity which implies that with any initial conditions the patterns that it produces can be obtained as simple superpositions of the first pattern shown above.
It turns out that they are both additive rules, implying that the patterns they produce can be superimposed in the way we discussed on page 264 .
At any point one can follow the arrow to the left to get a black cell, but the form of the network implies that this black cell must always be followed by at least one white cell.
And as a trivial example, the constraint in a line of black or white cells that every cell should be the same color as both its neighbors immediately implies that the whole line must be either uniformly black or uniformly white.
And what we see is that the entropy does indeed tend to increase, just as the Second Law implies.
So in particular this implies that it should be possible to think of processes in nature as computations.
For the completeness of the system guarantees that one must always reach one or the other, while its consistency implies that reaching one allows one to conclude that one will never reach the other.
It so happens that some of these statements then imply intrinsic general limitations on science—but even the very fact that such statements can be made is in a sense an example of successful generality in science that goes against the conclusions of postmodernism.
And the axioms then turn out to imply that any additional objects must be larger than all these integers—and must therefore be infinite.