hard to believe that systems this simple could ever be universal, and thus in a sense be able to emulate essentially any system.
But from the discoveries in this book this now seems almost inevitable. And indeed the Principle of Computational Equivalence implies that beyond some low threshold almost any axiom system should be expected to be universal.
So how does universality actually work in the case of arithmetic?
One approach is illustrated in the picture on the next page. The idea is to set up an arithmetic statement that can be proved true if the evolution of a cellular automaton from a given initial condition makes a given cell be a given color at a given step, and can be proved false if it does not.
By changing numbers in this arithmetic statement one can then in effect sample different aspects of the cellular automaton evolution. And with the cellular automaton being a universal one such as rule 110 this implies that the axioms of arithmetic can support universality.
Such universality then implies Gödel's Theorem and shows that there must exist statements about arithmetic that cannot ever be proved true or false from its normal axioms.
So what are some examples of such statements?
The original proof of Gödel's Theorem was based on considering the particular self-referential statement "this statement is unprovable".
At first it does not seem obvious that such a statement could ever be set up as a statement in arithmetic. But if it could then one can see that it would immediately follow that—as the statement says—it cannot be proved, since otherwise there would be an inconsistency.
And in fact the main technical difficulty in the original proof of Gödel's Theorem had to do with showing—by doing what amounted to establishing the universality of arithmetic—that the statement could indeed meaningfully be encoded as a statement purely in arithmetic.
But at least with the original encoding used, the statement would be astronomically long if written out in the notation of page 773. And from this result, one might imagine that unprovability would never be relevant in any practical situation in mathematics.
But does one really need to have such a complicated statement in order for it to be unprovable from the axioms of arithmetic?