Computability and Logic

226 INDEFINABILITY, UNDECIDABILITY, INCOMPLETENESS
It follows by pure logic that
⊢ T ∀y(Prf T ( R T , y) →∃z < y Disprf( R T , z))
and hence ⊢ T R T , and T is inconsistent, a contradiction that shows that R T cannot be
disprovable in T .
A theory T is called ω-inconsistent if and only if for some formula F(x), ⊢ T ∃xF(x)
but ⊢ T ∼F(n) for every natural number n, and is called ω-consistent if and only if
it is not ω-inconsistent. Thus an ω-inconsistent theory ‘affirms’ there is some number
with the property expressed by F, but then ‘denies’ that zero is such a number,
that one is such a number, than two is such a number, and so on. Since ∃xF(x) and
∼F(0), ∼F(1), ∼F(2),...cannot all be correct, any ω-inconsistent theory must have
some incorrect theorems. But an ω-inconsistent theory need not be inconsistent. (An
example of a consistent but ω-inconsistent theory will be given shortly.)
17.9 Theorem. Let T be a consistent, axiomatizable extension of Q. Then a Gödel
sentence for T is unprovable in T , and if T is ω-consistent, it is also undisprovable in T .
Proof: Suppose the Gödel sentence G T is provable in T . Then the ∃-rudimentary
sentence ∃y Prf T ( G T , y) is correct, and so provable in T . But since G T is a Gödel
sentence, G T ↔∼∃y Prf T ( G T , y) is also provable in T . By pure logic it follows
that ∼G T is provable in T , and T is inconsistent, a contradiction, which shows that
G T is not provable in T .
Suppose the sentence G T is disprovable in T . Then ∼∼∃y Prf T ( G T , y) and hence
∃y Prf T ( G T , y) is provable in T . But by consistency, G T is not provable in T , and so
for any n, n is not a witness to the provability of G T , and so the rudimentary sentence
∼Prf T ( G T , n) is correct and hence provable in Q and hence in T . But this means
T is ω-inconsistent, a contradiction, which shows that G T is not disprovable in T .
For an example of a consistent but ω-inconsistent theory, consider the theory
T = Q +∼G Q consisting of all consequences of the axioms of Q together with ∼G Q
or ∼∼∃y Prf Q (G Q , y). Since G Q is not a theorem of Q, this theory T is consistent. Of
course ∃y Prf Q (G Q , y) is a theorem of T . But for any particular n, the rudimentary
sentence ∼Prf Q (G Q , n) is correct, and therefore provable in any extension of Q,
including T .
Historically, Theorem 17.9 came first, and Theorem 17.8 was a subsequent refinement.
Accordingly, the Rosser sentence is sometimes called the Gödel–Rosser
sentence. Subsequently, many other examples of undecidable sentences have been
brought forward. Several interesting examples will be discussed in the following,
optional section, and the most important example in the next chapter.
17.3* Undecidable Sentences without the Diagonal Lemma
The diagonal lemma, which was used to construct the Gödel and Rosser sentences,
is in some sense the cleverest idea in the proof of the first incompleteness theorem,
and is heavily emphasized in popularized accounts. However, the possibility of implementing
the idea of this lemma, of constructing a sentence that says of itself that
it is unprovable, depends on the apparatus of the arithmetization of syntax and the