Computability and Logic

17.3*. UNDECIDABLE SENTENCES WITHOUT THE DIAGONAL LEMMA 229
of symbols in any encyclopedia remains far less than 10 ⇑ 10. So if n is denominated
by the formula L(x), its complexity is less than 10 ⇑ 10. Since this is impossible, it
follows that no sentence of form GC(n) can be proved: no specific number n can be
proved to have complexity greater than 10 ⇑ 10. This reasoning can be adapted to
any other reasonable measure of complexity.
(For example, suppose we take the complexity of a number to be the smallest
number of states needed for a Turing machine that will produce that number as output
given zero as input. To establish that ‘the complexity of x is y’ and related formulas
can be expressed in the language of arithmetic we now need the fact that Turing
machines can be coded by recursive functions in addition to the fact that recursive
functions are representable. And to show that if there is any proof that some number
has complexity greater than 10 ⇑ 10, then the number n identified by the lead witness
can be generated as the output for input zero by some Turing machine, we need in
addition to the arithmetizability of syntax the fact also of the Turing computability of
recursive functions. Almost the whole of this book up to this point is involved just in
outlining how one would go about writing down the relevant formula and designing
the relevant Turing machine. But while filling in the details of this outline might fill
an encyclopedia, still it would not require anything approaching 10 ⇑ 10 symbols,
and that is all that is essential to the argument. In the literature, the label Chaitin’s
theorem refers especially to this Turing-machine version, but as we have said, similar
reasoning applies to any reasonable notion of complexity.)
Thus on any reasonable measure of complexity, there is an upper bound b—we
have used 10 ⇑ 10, though a closer analysis would show that a much smaller number
would do, its exact value depending on the particular measure of complexity being
used—such that no specific number n can be proved in Q to have complexity greater
than b. Moreover, this applies not just to Q but to any stronger true theory, such as
P or the theories developed in works on set theory that are adequate for formalizing
essentially all ordinary mathematical proofs. Thus Chaitin’s theorem, whose proof we
have sketched, tells us that there is an upper bound such that no specific number can be
proved by ordinary mathematical means to have complexity greater than that bound.
Problems
17.1 Show that the existence of a semirecursive set that is not recursive implies
that any consistent, axiomatizable extension of Q fails to prove some correct
∀-rudimentary sentence.
17.2 Let T be a consistent, axiomatizable theory extending Q. Consider the set P yes
of (code numbers of) formulas that are provable in T , and the set P no of (code
numbers of) formulas that are disprovable in P. Show that there is no recursive
set R such that P yes is a subset of R while no element of R is an element of P no .
17.3 Let B 1 (y) and B 2 (y) be two formulas of the language of arithmetic. Generalizing
the diagonal lemma, show that there are sentences G 1 and G 2 such that
⊢ Q G 1 ↔ B 2 ( G 2 )
⊢ Q G 2 ↔ B 1 ( G 1 ).