Download course notes - Theory and Logic Group

2.4
Incompleteness
We will now make precise what is meant by a syntactic characterization of the set of arithmetically true sentences.
Definition 2.15. A theory T is called axiomatisable if there is a recursive set Γ
T ClpΓq.
„
T s.t.
The above recursive set Γ can best be thought of as a set of axioms for T which motivates the
choice of the term axiomatisable. Being axiomatisable is a property which we require in order
to accept a characterisation of a theory as a syntactic one.
Lemma 2.4. If T is an axiomatisable theory then it is recursively enumerable.
Proof. Let T ClpΓq and Γ be recursive. Let A0 , A1 , A2 , . . . be an enumeration of all valid sentences in the language of arithmetic which exists by the abstract completeness theorem (Theorem 2.2). The following algorithm enumerates T :
Input: i P N
if Ai is of the form B1 ^ . . . ^ Bk Ñ C and Bj P Γ for j 1, . . . , k then
print C
else
print D
end if
where D is any sentence provable in T .
Lemma 2.5. If T is an axiomatisable and complete theory then it is recursive.
Proof. If T is inconsistent, it is the set of all sentences which is recursive. So assume T is
consistent. Then, as T is also complete, T & A iff T $ A. Let A0 , A1 , A2 , . . . be a recursive
enumeration of T which exists by Lemma 2.4. The following algorithm is a decision procedure
for T :
Input: formula A
kÐ0
loop
if Ak A then
return 1
end if
if Ak A then
return 0
end if
end loop
By completeness of T the above algorithm terminates for every input A.
The kernel of the proof of the first incompleteness theorem is a diagonal argument which we
first formulate in a general, abstract form. Let R „ N2 , for m P N we write Rm for tn P N |
pm, nq P Ru.
Lemma 2.6 (Diagonal Lemma). If R „ N2 and P
m.
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t
n P N | pn, nq R Ru then P
Rm for all
Proof. Suppose P
Lemma 2.7. If T
…
Rm , then pm, mq P R iff m P Rm iff m P P iff pm, mq R R.
Q is a consistent theory, then T is not recursive.
Proof. We fix a variable z. For a formula A define
E pAq : tn P N | T
For every recursive set S
theorem, i.e.
1. n P S implies T
$
2. n R S implies T
n R E pAq.
Therefore S
„
$
ssu
Arz zn
N there is a formula A which represents S in T by the representability
ss hence n P E pAq and
Arz zn
$
ss which, by consistency of T , implies T
Arz zn
&
ss hence
Arz zn
E pAq. Let now
R : tpm, nq P N2 | m #F, T
$
ssu
F rz zn
and observe that
R#A
nPN|T
t
$
ssu E pAq.
Arz zn
Furthermore, let
P : tn P N | pn, nq R Ru
n P N | n #F or n #F, T
t
&
F rz zn
ssu.
By the diagonal lemma P Rm for all m so, in particular, P R#A for all A so P E pAq
for all A. But every recursive set is representable in Q hence in T therefore P is not recursive.
Then also T is not recursive for suppose it would be, then also P would be recursive.
Theorem 2.3 (First Incompleteness Theorem). If T
theory, then T is not complete.
…
Q is a consistent and axiomatisable
Proof. Suppose T … Q is consistent, axiomatisable and complete. By consistency and Lemma 2.7
it is not recursive. But by axiomatizability and completeness and Lemma 2.5 it is recursive.
Contradiction.
Corollary 2.1. ThpN q is not axiomatisable.
Proof. ThpN q … Q because the axioms of Q are true in N . Also ThpN q is (as every theory of
a structure) consistent and complete.
Corollary 2.2 (Undecidability of First-Order Logic). The set of valid sentences is not recursive.
Proof. Let A be the conjunction of the axioms of Q. Suppose the set of valid sentences would
be recursive then also Q would be recursive because Q $ B iff A Ñ B is valid. But Q is not
recursive by the above Lemma 2.7.
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The set of valid sentences is therefore an example for a set which is recursively enumerable but
not recursive. There are more direct proofs of the undecidability of first-order logic which do
not use the first incompleteness theorem.
A sentence A is called undecidable in a theory T if T & A and T & A. Be careful to not
confuse this notion with that of a decidable set (which in our terminology is a recursive set).
The first incompleteness theorem shows that every consistent and axiomatisable extension of
Q has an undecidable sentence. However, the above proof does not directly give an example of
such a sentence. We will now carry out an alternative proof by constructing such a sentence
explicitely. This other proof will also be useful for the second incompleteness theorem. Before
that we need some preparatory steps:
Definition 2.16. A bounded quantifier is a quantifier of the form Dx px
t Ñ B q.
t ^ B q or @x px
A formula A is called Σ1 -formula if it is logically equivalent to a formula of the form Dx1 Dxn B
where all quantifiers in B are bounded.
A formula A is called Π1 -formula if it is logically equivalent to a formula of the form @x1 @xn B
where all quantifiers in B are bounded.
Lemma 2.8 (Σ1 -completeness of Q). If A is a Σ1 -sentence with N
(
A, then Q $ A.
Without Proof.
Lemma 2.9. There is a Σ1 -formula Numpx, y q s.t. for all k, n P N with n #s
k:
Q $ @y pNumps
k, y q Ø y
n
sq
There is a Σ1 -formula Subpx1 , x2 , x3 , x4 q s.t. for all a, v, t, b P N where a
A, v #z for a variable z, t #s for a term s and b #Arz zts:
a, vs, ss, y q Ø y
Q $ @y pSubps
#A for a formula
sbq
If T is an axiomatisable theory, then there is a Σ1 -formula ProvT pxq s.t. for all n P N
sq
Q $ ProvT pn
n #A and T
iff
$
A
for some formula A.
Proof Sketch. The corresponding functions and predicate are recursive.
We will use Gödel-numerals, these are the numerals of Gödel-numbers. More precisely, for an
„
expression (i.e. a term, a formula,...) e, the Gödel-numeral of e is defined as xey : #e.
Example 2.6. 0 P N and s
0 0 is also an LN -term. x0y is again an LN -term, but x0y 0 because
„ #p‡
x0y #0
2, 0, 0q and #p2, 0, 0q P N and #p2, 0, 0q 0.
For any expression e: Q $ @y pNumpxey, y q Ø y
xxeyyq.
We can now give an alternative proof of a slightly modified version of the first incompleteness
theorem.
Theorem 2.4. If T is a consistent and axiomatisable theory with ThpN q
incomplete.
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…
T
…
Q then T is
In the proof of Lemma 2.7 we have used sets R P N2 and P
#A P P
iff
p
#A, #Aq R R
iff
T
P
N s.t.
&
Arz zxAys
Proof. Now we will formalise the above condition inside the language of arithmetic. To that
aim define the formula
B pz q : D
uDv pNumpz, uq ^ Subpz, xz y, u, v q ^ ProvT pv qq.
which exists as T is axiomatisable. Let us define the so-called Gödel-sentence G : B pxB yq
which expresses that G is not provable in T because:
N
(
G
iff
N
iff
T
&
B rz zxB ys
iff
T
&
G
* D
uDv pNumpxB y, uq ^ SubpxB y, xz y, u, v q ^ ProvT pv qq
Suppose N * G, then by the above T $ G. Furthermore, G is a Π1 -sentence, so G is a
Σ1 -sentence and by Σ1 -completeness we know that N ( G implies T $ G and T would be
inconsistent, contradiction, so
N ( G.
Suppose now T
$
G, then N
(
G, contradiction, so
T
G.
&
Suppose T $ G then N * G by the above so N ( G and by Σ1 -completeness T
contradicts consistency, so
T & G.
$
G which
The above sentence G of a theory T is therefore true but undecidable in T . The second
incompleteness theorem will give another example of a sentence which is undecidable in T : the
sentence which expresses the consistency of T
ConT :
ProvT pxKyq.
The second incompleteness theorem applies to arithmetical theories which are stronger than
minimal arithmetic. The reason for this is that the proof of the second incompleteness theorem
is essentially a formalisation in T of the above alternative proof of the first incompleteness
theorem about T .
Definition 2.17. The arithmetical theory of Peano Arithmetic (PA) is defined as the deductive
closure of Q and all sentences of the form
Ap0q Ñ @x pApxq Ñ Apspxqqq Ñ @x Apxq,
the induction axioms.
Theorem 2.5 (Second Incompleteness Theorem). If T
then T & ConT .
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…
PA is consistent and axiomatisable,
Proof. Let G be the Gödel-sentence for T . Then
T
G Ñ ProvT px Gyq
$
because PA proves the Σ1 -completeness of Q. On the other hand
G
D
uDv pNumpxB y, uq ^ SubpxB y, xz y, u, v q ^ ProvT pv qq
and by Σ1 -completeness
T
$
NumpxB y, xxB yyq
and
T
$
SubpxB y, xz y, xxB yy, xGyq,
hence using representability of the Num- and Sub-functions
T
G Ñ ProvT pxGyq.
$
Furthermore, T proves that ProvT is closed under the inference rules of NK so
T
G Ñ ProvT pxKyq,
$
i.e.
T
$
ConT
Ñ
G
but as we have seen in the alternative proof of the first incompleteness theorem
T
&
G
&
ConT .
and therefore
T
In order to fully appreciate the importance of the incompleteness theorems of Gödel it is necessary to say a few words about the historical context. Gödel’s work is a reaction to Hilbert’s
programme, which called for a formalisation of mathematical reasoning and for a proof of the
consistency of this formalism by “finitary” means. It has never been made completely precise
what should be understood as “finitary” but the intention of providing such a proof was to
obtain an absolute justification of mathematical reasoning. So the methods employed in this
consistency proof should themselves be undoubtable. The second incompleteness theorem shows
that this program cannot be fulfilled: for proving the consistency of a theory the theory must
be transcended.
Finally one should discuss the mathematical significance of the incompleteness theorems. The
examples of undecidable sentences we have seen so far, the Gödel-sentence of a theory as well
as the consistency of a theory, are purely logical statements. One might think at first sight that
this incompleteness phenomenon only manifests itself on such self-referential statements which
are – outside of logic – hardly used in mathematics. However, this impression is not justified.
There are sentences of purely mathematical (as opposed to logical) nature that have been shown
to be independent of strong theories. The primary example is the continuum hypothesis, the
first of Hilbert’s famous 23 problems. We again work in the language of set theory; the linear
ordering of sets by comparing their cardinality is given by
x ¤ y :
D
ϕ pϕ : x Ñ y ^ injectivepϕqq
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and
x y : x ¤ y ^ y
¤
x
Then the continuum hypothesis is the statement
CH
D
x p N x Rq
which has been shown to be independent of Zermelo-Fraenkel set theory ZFC by Gödel and
Cohen.
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