Computability and Logic

228 INDEFINABILITY, UNDECIDABILITY, INCOMPLETENESS
k symbols. For logically equivalent formulas denominate the same number (if they
denominate any number at all), and every formula with fewer than k symbols is logically
equivalent, by relettering bound variables, to one with only the first k variables
on our official list of variables, and there are only finitely many of those.
Thus, there will be numbers not denominable using fewer than 10 ⇑ 10 symbols.
The usual apparatus allows us to construct a Gödel–Berry formula GB(x, y), expressing
‘x is the least number not denominable by a formula with fewer than y ⇑ y
symbols’. Writing out this formula would involve writing out not just the formula
representing the super-exponential function ⇑, but also the formulas relating to provability
in Q. Again we have not actually written out these formulas, but only given
an outline of how to do so in our proofs of the arithmetizability of syntax and the
representability of recursive functions in Q. Review of those proofs reveals that writing
out the formula GB(x, y) orGB(x, 10), though it would require more time and
paper than any reasonable homework assignment, would not require more symbols
than appear in an ordinary encyclopedia, which is far fewer than the astronomical
figure 10 ⇑ 10. Now there is some number not denominable by a formula with fewer
symbols than that astronomical figure, and among such numbers there is one and
only one least, call it n. Then GB(n, 10) and ∀x(GB(x, 10) ↔ x = n) are true. But
if the latter were provable, the formula GB(x, 10) would denominate n, whereas n is
not denominable except by formulas much longer than that. Hence we have another
example of an unprovable truth.
This example is worth pressing a little further. The length of the shortest formula
denominating a number may be taken as a measure of the complexity of that number.
Just as we could construct the Gödel–Berry formula, we can construct a formula
C(x, y, z) expressing ‘the complexity of x is y and y is greater than z ⇑ z’, and
using it the Gödel–Chaitin formula GC(x) or∃yC(x, y, 10), expressing that x has
complexity greater than 10 ⇑ 10. Now for all but finitely many n, GC(n) is true.
Chaitin’s theorem tells us that no sentence of form GC(n) is provable.
The reason may be sketched as follows. Just as ‘y is a witness to the provability
of x in Q’ can be expressed in the language of arithmetic by a formula Prf Q (x,y), so
can ‘y is a witness to the provability of the result of subsituting the numeral for x for
the variable in GC’ be expressed by a formula PrfGC Q (x,y). Now if any sentence of
form GC(n) can be proved, there is a least m such that m witnesses the provability of
GC(n) for some n. Let us call m the ‘lead witness’ for short. And of course, since any
one number witnesses the provability of at most one sentence, there will be a least
n—in fact, there will be one and only one n—such that the lead witness is a witness to
the provability of GC(n). Call n the number ‘identified by the lead witness’ for short.
If one is careful, one can arrange matters so that the sentences K (m) and L(n)
expressing ‘m is the lead witness’ and ‘n is the number identified by the lead witness’
will be ∃-rudimentary, so that, being true, K (m) and L(n) will be provable. Moreover,
since it can be proved in Q that there is at most one least number fulfilling the condition
expressed by any formula, ∀x(x ≠ m →∼K (x)) and ∀x(x ≠ n →∼L(x)) will also
be provable. But this means that n is denominated by the formula L(x), and hence
has complexity less than the number of symbols in that formula. And though it might
take an encyclopedia’s worth of paper and ink to write the formula down, the number