Computability and Logic

17.3*. UNDECIDABLE SENTENCES WITHOUT THE DIAGONAL LEMMA 227
representability of recursive functions. Once that apparatus is in place, a version of
the incompleteness theorem, showing the existence of a true but unprovable sentence,
can be established without the diagonal lemma. One way to do so is indicated in the
first problem at the end of this chapter. (This way uses the fact that there exist semirecursive
sets that are not recursive, and though it does not use the diagonal lemma,
does involve a diagonal argument, buried in the proof of the fact just cited.) Some
other ways will be indicated in the present section.
Towards describing one such way, recall the Epimenides or liar paradox, involving
the sentence ‘This sentence is untrue’. A contradiction arises when we ask whether
this sentence is true: it seems that it is if and only if it isn’t. The Gödel sentence
in effect results from this paradoxical sentence on substituting ‘provable’ for ‘true’
(a substitution that is crucial for establishing that we can actually construct a Gödel
sentence in the language of arithmetic). Now there are other semantic paradoxes,
paradoxes in the same family as the liar paradox, involving other semantic notions
related to truth. One famous one is the Grelling or heterological paradox. Call an
adjective autological if it is true of itself, as ‘short’ is short, ‘polysyllabic’ is polysyllabic,
and ‘English’ is English, and call it heterological if it is untrue of itself, as
‘long’ is not long, ‘monosyllabic’ is not monosyllabic, and ‘French’ is not French. A
contradiction arises when we ask whether ‘heterological’ is heterological: it seems
that it is if and only if it isn’t.
Let us modify the definition of heterologicality by substituting ‘provable’ for
‘true’. We then get the notion of self-applicability: a number m is self-applicable
in Q if it is the Gödel number of a formula μ(x) such that μ(m) is provable in Q.
Now the same apparatus that allowed us to construct the Gödel sentence allows us
to construct what may be called the Gödel–Grelling formula GG(x) expressing ‘x
is not self-applicable’. Let m be its Gödel number. If m were self-applicable, then
GG(m) would be provable, hence true, and since what it expresses is that m is not
self-applicable, this is impossible. So m is not self-applicable, and hence GG(m) is
true but unprovable.
Another semantic paradox, Berry’s paradox, concerns the least integer not namable
in fewer than nineteen syllables. The paradox, of course, is that the integer in question
appears to have been named just now in eighteen syllables. This paradox, too, can be
adapted to give an example of a sentence undecidable in Q. Let us say that a number
n is denominable in Q by a formula φ(x) if∀x(φ(x) ↔ x = n) is (not just true but)
provable in Q.
Every number n is denominable in Q, since if worse comes to worst, it can always
be denominated by the formula x = n, a formula with n + 3 symbols. Some numbers
n are denominable in Q by formulas with far fewer than n symbols. For example, the
number 10 ⇑ 10 is denominable by the formula φ(10, 10, x), where φ is a formula
representing the super-exponential function ⇑. We have not actually written out this
formula, but instructions for doing so are implicit in the proof that all recursive functions
are representable, and review of that proof reveals that writing out the formula
would not take more time or more paper than an ordinary homework assignment. By
contrast, 10 ⇑ 10 is larger than the number of particles in the visible universe. But
while big numbers can thus be denominated by comparatively short formulas, for any
fixed k, only finitely many numbers can be denominated by formulas with fewer than