Chapter 12: The Principle of Computational Equivalence

Section 3: The Content of the Principle

Density of universal systems

One might imagine that it would be possible to make estimates of the overall density of universal systems, perhaps using arguments like those for the density of primes, or for the density of algorithmically random sequences. But as it turns out I know of no way to make any such estimates. If one has shown that various simple rules are universal, then it follows that rules which generalize these must also be universal. But even from this I do not know, for example, how to prove that the density of universal rules cannot decrease when rules become more complicated.

From Stephen Wolfram: A New Kind of Science [citation]