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CHAPTER 4
Gödel’s Theorems
4.1. Gödel Numbers
We will assign numbers – so-called Gödel numbers, GN for short – to the
syntactical constructs developed in chapter 1: terms, formulas and derivations. Using the elementary sequence-coding and decoding machinery developed earlier we will be able to construct the code number of a composed
object from its parts, and conversely to disassemble the code number of a
composed object into the code numbers of its parts.
4.1.1. Gödel numbers of terms, formulas and derivations. Let
L be a countable first order language. Assume that we have injectively
assigned to every n-ary relation symbol R a symbol number SN(R) of the
form h1, n, ii and to every n-ary function symbol f a symbol number SN(f ) of
the form h2, n, ji. Call L elementarily presented if the set SymbL of all these
symbol numbers is elementary. In what follows we shall always assume that
the languages L considered are elementarily presented. In particular this
applies to every language with finitely many relation and function symbols.
Let SN(Var) := h0i. For every L-term r we define recursively its Gödel
number prq by
pxi q
:= hSN(Var), ii,
pf r1 . . . rn q := hSN(f ), pr1 q, . . . , prn qi.
Assign numbers to the logical symbols by SN(→) := h3, 0i und SN(∀) :=
h3, 1i. For simplicity we leave out the logical connectives ∧, ∨ and ∃ here;
they could be treated similarly. We define for every L-formula A its Gödel
number pAq by
pRr1 . . . rn q := hSN(R), pr1 q, . . . , prn qi,
pA → Bq
:= hSN(→), pAq, pBqi,
p∀xi Aq
:= hSN(∀), i, pAqi.
We define symbol numbers for the names of the natural deduction rules:
SN(AVar) := h4, 0i, SN(→+ ) := h4, 1i, SN(→− ) := h4, 2i, SN(∀+ ) := h4, 3i,
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4. GÖDEL’S THEOREMS
SN(∀− ) := h4, 4i. For a derivation M we define its Gödel number pM q by
puA
i q
:= hSN(AVar), i, pAqi,
pλuA M q := hSN(→+ ), i, pAq, pM qi,
i
pM N q
:= hSN(→− ), pM q, pN qi,
pλxi M q := hSN(∀+ ), i, pM qi,
pM rq
:= hSN(∀− ), pM q, prqi.
4.1.2. Elementary functions on Gödel numbers. We shall define
an elementary predicate Deriv such that Deriv(d) if and only if d is the
Gödel number of a derivation. To this end we need a number of auxiliary
functions and relations, which will all be elementary and have the properties
formulated below. First we need to provide some notions.
Ter(t)
t is GN of a term,
For(a)
a is GN of a formula,
FV(i, y)
the variable xi is free in the term or formula with GN y,
fla(d)
GN of the formula derived by the derivation with GN d.
A
n−1
0
By the context of a derivation M we mean the set {uA
i0 , . . . , uin−1 } of its free
assumption variables, where i0 < · · · < in−1 . Its Gödel number is defined to
be the least number c such that ∀ν<n ((c)iν = pAi q).
ctx(d)
GN of the context of the derivation with GN d,
Cons(c1 , c2 ) the contexts with GN c1 , c2 are consistent.
Then Deriv can be defined by course-of-values recursion:
Deriv(d) := ((d)0 = SN(AVar) ∧ lh(d) = 3 ∧ For((d)2 )) ∨
((d)0 = SN(→+ ) ∧ lh(d) = 4 ∧ For((d)2 ) ∧ Deriv((d)3 ) ∧
((ctx((d)3 ))(d)1 6= 0 → (ctx((d)3 ))(d)1 = (d)2 )) ∨
((d)0 = SN(→− ) ∧ lh(d) = 3 ∧ Deriv((d)1 ) ∧ Deriv((d)2 ) ∧
Cons(ctx((d)1 ), ctx((d)2 )) ∧
(fla((d)1 ))0 = SN(→) ∧ (fla((d)1 ))1 = fla((d)2 )) ∨
((d)0 = SN(∀+ ) ∧ lh(d) = 3 ∧ Deriv((d)2 ) ∧ ∀i<lh(ctx((d)2 )) (
(ctx((d)2 ))i 6= 0 → ¬FV((d)1 , (ctx((d)2 ))i ))) ∨
((d)0 = SN(∀− ) ∧ lh(d) = 3 ∧ Deriv((d)1 ) ∧ Ter((d)2 ) ∧
(fla((d)1 )0 = SN(∀)).
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For to define the auxiliary functions and relations we need further notions.
A substitution is a map xi0 7→ r0 , . . . , xin−1 7→ rn−1 with i0 < · · · < in−1
from variables to terms; its Gödel number is the least number s such that
∀ν<n ((s)iν = pri q).
union(c1 , c2 )
GN of the union of the consistent contexts with GN c1 , c2 ,
remove(c, i)
GN of result of removing ui from the context with GN c,
Sub(x, s, y)
y is the GN of the result of applying the substitution
with GN s to the term or formula with GN x,
update(s, i, t)
GN of the result of updating the substitution with GN s
by changing its entry at i to the term with GN t,
We now give definitions of all these; from the form of the definitions it will
be clear that they have the required properties, and that the relations and
functions thus defined are elementary.
update(s, i, t) :=
µx<h(max(s,t),max(lh(s),i)) ((x)i = t ∧ ∀k<max(lh(s),i) (k 6= i → (x)i = (s)i )).
k
Here h(n, k) := (n + 1)2 is the elementary function defined earlier with the
property
hn, . . . , ni ≤ h(n, k).
| {z }
k
We define substitution by
Sub(x, s, y) :=
((x)0 = SN(Var) ∧ (s)(x)1 = 0 ∧ y = x) ∨
((x)0 = SN(Var) ∧ (s)(x)1 6= 0 ∧ y = (s)(x)1 ) ∨
(((x)0,0 = 1 ∨ (x)0,0 = 2 ∨ (x)0 = SN(→)) ∧
lh(x) = lh(y) ∧ (x)0 = (y)0 ∧ ∀k<lh(x)−·1 ((x)i+1 = (y)i+1 )) ∨
((x)0 = SN(∀) ∧ ∀i<lh(s) ((s)i 6= 0 → ¬FV((x)1 , (s)i )) ∧
lh(x) = lh(y) ∧ (y)0 = SN(∀) ∧ (y)1 = (x)1 ∧ Sub((x)2 , s, (y)2 )) ∨
((x)0 = SN(∀) ∧ ∃i<lh(s) ((s)i 6= 0 ∧ FV((x)1 , (s)i )) ∧
lh(x) = lh(y) ∧ (y)0 = SN(∀) ∧ (y)1 = x ∧
Sub((x)2 , update(s, (x)1 , hSN(Var), xi), (y)2 )).
Using the lemma in 4.1.3 below one can see that, if x is GN of a term or a
x+s
formula and s is GN of a substitution, Sub(x, s, y) → y ≤ (x + s)2 .
Removal of an assumption variable from a context is defined by
remove(c, i) := µx≤c ((x)i = 0 ∧ ∀j<lh(c) (j 6= i → (x)j = (c)j )).
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4. GÖDEL’S THEOREMS
The union of two consistent contexts can again be defined by the bounded
µ-operator:
union(c1 , c2 ) := µc≤c1∗c2 ∀i<max(lh(c1 ),lh(c2 )) ((c)i = max((c1 )i , (c2 )i )).
Consistency of two contexts is defined by
Cons(c1 , c2 ) := ∀i<max(lh(c1 ),lh(c2 )) ((c1 )i 6= 0 → (c2 )i 6= 0 → (c1 )i = (c2 )i ).
The context of a derivation is defined by
ctx(d) := µc≤d (((d)0 = SN(AVar) ∧ (c)(d)1 = (d)2 ) ∨
((d)0 = SN(→+ ) ∧ c = remove(ctx((d)1 ), i)) ∨
((d)0 = SN(→− ) ∧ c = union(ctx((d)1 ), ctx((d)2 ))) ∨
((d)0 = SN(∀+ ) ∧ c = ctx((d)2 )) ∨
((d)0 = SN(∀− ) ∧ c = ctx((d)1 ))).
The formula of a derivation is defined by
fla(d) := µa≤d2d (((d)0 = SN(AVar) ∧ a = (d)2 ) ∨
((d)0 = SN(→+ ) ∧ a = hSN(→), (d)2 , fla((d)3 )i) ∨
((d)0 = SN(→− ) ∧ a = (fla((d)1 ))2 ) ∨
((d)0 = SN(∀+ ) ∧ a = hSN(∀), (d)1 , fla((d)2 )i) ∨
((d)0 = SN(∀− ) ∧
Sub((fla((d)1 ))2 , µs≤d ((s)(fla((d)1 ))1 = (d)2 ), a))).
Freeness of a variable xi in a term or formula is defined by
FV(i, y) := ((y)0 = SN(Var) ∧ (y)1 = i) ∨
((y)0,0 = 1 ∧ ∃j<lh(y)−·1 FV(i, (y)j+1 )) ∨
((y)0,0 = 2 ∧ ∃j<lh(y)−·1 FV(i, (y)j+1 )) ∨
((y)0 = SN(→) ∧ (FV(i, (y)1 ) ∨ FV(i, (y)2 ))) ∨
((y)0 = SN(∀) ∧ i 6= (y)1 ∧ FV(i, (y)2 )).
The sets of formulas and terms are defined by
For(a) :=
((a)0,0 = 1 ∧ SymbL ((a)0 ) ∧ lh(a) = (a)0,1 + 1 ∧ ∀j<(a)0,1 Ter((a)j+1 )) ∨
((a)0 = SN(→) ∧ lh(a) = 3 ∧ For((a)1 ) ∧ For((a)2 )) ∨
((a)0 = SN(∀) ∧ lh(a) = 3 ∧ For((a)2 )),
Ter(t) := ((t)0 = SN(Var) ∧ lh(t) = 2) ∨
((t)0,0 = 2 ∧ SymbL ((t)0 ) ∧ lh(t) = (t)0,1 +1 ∧ ∀j<(t)0,1 Ter((t)j+1 )).
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Recall that for simplicity we have left out the logical connectives ∧, ∨ and
∃. They could be added easily, including an extension of the notion of a
derivation to also allow their axioms as listed in 1.2.3.
4.1.3. Substitution of numerals. In the sequel the elementary relation Sub representing substitution will not suffice. We shall need an elementary function s such that s(pC(z)q, k) = pC(k)q where k denotes the
numeral k for the number k, defined by 0 := 0 and k + 1 := Sk (in any
language with the function symbols 0 and S). It is easy to write down the
course-of-values recursion equations for s; however, we also need to find an
elementary bounding function. We shall do this for terms r(z); the argument
can easily be extended to formulas C(z).
For a term r we define its sum of maximal sequence lengths ||r|| by
||xi || := 2
and ||f r0 . . . rk−1 || := k + 1 + max(||ri ||)
i<k
and its symbol bound Symb(r) by
Symb(xi ) := max(SN(Var), i) + 1
Symb(f r0 . . . rk−1 ) := max(SN(f ), max(Symb(ri ))).
i<k
2||r||
Lemma. ||r|| ≤ prq < Symb(r)
.
Proof. We prove the first inequality by induction on r. The case of a
variable xi is easy:
||xi || = 2 ≤ hSN(Var), ii = pxi q,
and for a term f r0 . . . rk−1 we have by the estimate in 3.2.5
pf r0 . . . rk−1 q = hSN(f ), pr0 q, . . . , prk−1 qi
≥ (k + 1)(max(SN(f ), pr0 q, . . . , prk−1 qi) + 1)
> k + 1 + max(pri q)
i<k
≥ k + 1 + max(||ri ||) by induction hypothesis
i<k
= ||f r0 . . . rk−1 ||.
For the second inequality we again use induction on k. For a variable xi we
obtain again by the estimate in 3.2.5
2
||xi ||
pxi q = hSN(Var), ii < Symb(xi )2 = Symb(xi )2
and for a term built with a function symbol f we have
pf r0 . . . rk−1 q
= hSN(f ), pr0 q, . . . , prk−1 qi
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4. GÖDEL’S THEOREMS
maxi<k ||ri ||
· 1, n −
· 1, . . . , n −
· 1i with n := Symb(r)2
≤ hn −
|
{z
}
, by ind. hyp.
k+1
<n
2k+1
by the estimate in 3.2.5
k+1+maxi<k ||ri ||
= Symb(r)2
||f r0 ...rk−1 ||
= Symb(r)2
.
As a consequence we obtain the desired estimates. First notice that
2k+1
pkq < c2
with c := max(SN(S), SN(0)), since by the previous lemma
||k||
2
pkq < c , and ||k|| = 2k + 1 can be proved easily by induction on k:
||0|| = 1, and ||Sk|| = 2 + ||k||. Now we obtain for z := x0 , again by the
previous lemma
||r(k)||
s(pr(z)q, k) = pr(k)q < max(Symb(r), SN(S), SN(0))2
||r(z)||+2k+1
≤ max(pr(z)q, SN(S), SN(0))2
pr(z)q+2k+1
≤ max(pr(z)q, SN(S), SN(0))2
.
Therefore we can bound the substitution function s for numerals by
y+2k+1
s(y, k) ≤ max(y, SN(S), SN(0))2
.
4.1.4. Axiomatized theories. Call a relation recursive if its (total)
characteristic function is recursive. A set S of formulas is called recursive
(elementary, recursively enumerable), if pSq := { pAq | A ∈ S } is recursive
(elementary, recursively enumerable). Clearly the sets StabL of stability
axioms and EqL of L-equality axioms are elementary. Now let L be an
elementarily presented language with = in L. A theory T with L(T ) ⊆ L is
recursively (elementarily) axiomatizable, if there is a recursive (elementary)
set S of closed L-formulas such that T = { A ∈ L | S ∪ EqL ⊢ A }.
Theorem. For theories T with L(T ) ⊆ L the following are equivalent.
(a) T is recursively axiomatizable.
(b) T is elementarily axiomatizable.
(c) T is recursively enumerable.
Proof. (c) → (b). Let pT q be recursively enumerable. Then there is
an elementary f such that pT q = ran(f ). Let f (n) = pAn q. We define an
elementary function g with the property g(n) = pA0 ∧ · · · ∧ An q by
g(0)
:= f (0),
g(n + 1) := g(n) ∧˙ f (n + 1),
g(n)
4n+m
≤ max(SN(∧), m)2
where m := maxi<n f (i)
4.1. GÖDEL NUMBERS
75
with a ∧˙ b := hSN(∧), a, bi. For S := { A0 ∧ · · · ∧ An | n ∈ N } we have pSq =
ran(g), and this set is elementary because of a ∈ ran(g) ↔ ∃n<a (a = g(n)).
T is elementarily axiomatizable, since T = { A ∈ L | S ∪ EqL ⊢ A }.
(b) → (a) is clear.
(a) → (c). Let T be axiomatized by S with pSq recursive. Then
a ∈ pT q ↔ ∃d (Deriv(d) ∧ fla(d) = a ∧ ∀i<a ¬FV(i, a) ∧
∀i<lh(ctx(d)) ((ctx(d))i ∈ pEqq ∪ pSq)).
Hence pT q is recursively enumerable.
Call a theory T in our elementarily presented language L axiomatized if it
is given by a recursively enumerable axiom system AxT . By the theorem just
proved we can even assume that AxT is elementary. For such axiomatized
theories we define a binary relation Prf T by
Prf T (d, a) := Deriv(d) ∧ fla(d) = a ∧ ∀i<lh(ctx(d)) ((ctx(d))i ∈ pEqq ∪ pSq).
Clearly Prf T is elementary and Prf T (d, a) if and only if d is the GN of a
derivation of the formula with GN a from a context composed of equality
axioms and formulas from AxT . A theory T is consistent if ⊥ ∈
/ T ; otherwise
T is inconsistent. A theory T is complete if for every closed formula A we
have A ∈ T or A ∈
/ T , and incomplete otherwise.
Corollary. Let T be a consistent theory. If T is axiomatized and
complete, then T is recursive.
Proof. We define the characteristic function cpT q of pT q as follows.
cpT q(a) is 0 if ¬For(a) or ∃i<a FV(i, a). Otherwise it is defined by
cpT q(a) = (µx ((Prf T ((x)0 , a) ∧ (x)1 = 1) ∨ (Prf T ((x)0 , ¬a)
˙ ∧ (x)1 = 0)))1
with ¬a
˙ := hSN(→), a, SN(⊥)i. Completeness of T implies that cpT q is total,
and consistency that it indeed is the characteristic function of pT q.
4.1.5. Undefinability of the notion of truth. Let M be an Lstructure. A relation R ⊆ |M|n is called definable in M if there is an
L-formula A(x1 , . . . , xn ) such that
R = { (a1 , . . . , an ) ∈ |M|n | M |= A(x1 , . . . xn )[x1 := a1 , . . . , xn := an ] }.
We assume in this section that |M| = N, 0 is a constant in L and S is a
unary function symbol in L with 0M = 0 and SM (a) = a + 1. Recall that
for every a ∈ N the numeral a ∈ TerL is defined by 0 := 0 and n + 1 := Sn.
Observe that in this case the definability of R ⊆ Nn by A(x1 , . . . , xn ) is
equivalent to
R = { (a1 , . . . , an ) ∈ Nn | M |= A(a1 , . . . , an ) }.
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4. GÖDEL’S THEOREMS
Furthermore let L be an elementarily presented language. We will always
assume in this chapter that every elementary relation is definable in M. A
set S of formulas is called definable in M if pSq := { pAq | A ∈ S } is.
We shall show that already from these assumptions it follows that the
notion of truth for M, more precisely the set Th(M) of all closed formulas
valid in M, is undefinable in M. From this it will follow that the notion of
truth is in fact undecidable, for otherwise the set Th(M) would be recursive (Church’s Thesis), hence recursively enumerable, and hence definable,
because we have assumed already that all elementary relations are definable
in M and so their projections are definable also. For the proof we shall need
the following Fixed Point Lemma, which will be generalized in 4.2.2.
Lemma (Semantical Fixed Point Lemma). If every elementary relation
is definable in M, then for every L-formula B(z) we can find a closed Lformula A such that
M |= A
if and only if
M |= B(pAq).
Proof. Let s be the elementary function from 4.1.3 satisfying for every
formula C = C(z) (with z := x0 )
s(pCq, k) = pC(k)q,
hence in particular
s(pCq, pCq) = pC(pCq)q.
By assumption the graph Gs of s is definable in M, by As (x1 , x2 , x3 ) say.
Let
C := ∃x (B(x) ∧ As (z, z, x)), A := C(pCq),
and therefore
A = ∃x (B(x) ∧ As (pCq, pCq, x)).
Hence M |= A if and only if ∃a∈N ((M |= B(a)) ∧ a = pC(pCq)q), which is
the same as M |= B(pAq).
Theorem (Tarski’s Undefinability Theorem). Assume that every elementary relation is definable in M. Then Th(M) is undefinable in M,
hence in particular not recursively enumerable.
Proof. Assume that pTh(M)q is definable by BW (z). Then for all
closed formulas A
M |= A if and only if
M |= BW (pAq).
Now consider the formula ¬BW (z) and choose by the Fixed Point Lemma
a closed L-formula A such that
M |= A
if and only if
This contradicts the equivalence above.
M |= ¬BW (pAq).
4.2. THE NOTION OF TRUTH IN FORMAL THEORIES
77
We already have noticed that all recursively enumerable relations are
definable in M. Hence it follows that pTh(M)q cannot be recursively enumerable.
4.2. The Notion of Truth in Formal Theories
We now want to generalize the arguments of the previous section. There
we have made essential use of the notion of truth in a structure M, i.e., of
the relation M |= A. The set of all closed fromulas A such that M |= A has
been called the theory of M, denoted Th(M).
Now instead of Th(M) we shall start more generally from an arbitrary
theory T . We consider the question as to whether in T there is a notion of
truth (in the form of a truth formula B(z)), such that B(z) “means” that
z is “true”. A consequence is that we have to explain all the notions used
without referring to semantical concepts at all.
• z ranges over closed formulas (or sentences) A, or more precisely
over their Gödel numbers pAq.
• A “true” is to be replaced by T ⊢ A.
• C “equivalent” to D is to be replaced by T ⊢ C ↔ D.
Hence the question now is whether there is a truth formula B(z) such that
T ⊢ A ↔ B(pAq) for all sentences A. The result will be that this is impossible, under rather weak assumptions on the theory T . Technically, the issue
will be to replace the notion of definability by the notion of “representability” within a formal theory. We begin with a discussion of this notion.
In this section we assume that L is an elementarily presented language
with 0, S and = in L, and T an L-theory containing the equality axioms
EqL .
4.2.1. Representable relations and functions.
Definition. A relation R ⊆ Nn is representable in T if there is a formula
A(x1 , . . . , xn ) such that
T ⊢ A(a1 , . . . , an )
if (a1 , . . . , an ) ∈ R,
T ⊢ ¬A(a1 , . . . , an ) if (a1 , . . . , an ) ∈
/ R.
A function f : Nn → N is called representable in T if there is a formula
A(x1 , . . . , xn , y) representing the graph Gf ⊆ Nn+1 of f , i.e., such that
(4.1)
T ⊢ A(a1 , . . . , an , f (a1 , . . . , an )),
(4.2)
T ⊢ ¬A(a1 , . . . , an , c)
if c 6= f (a1 , . . . , an )
and such that in addition
(4.3) T ⊢ A(a1 , . . . , an , y) ∧ A(a1 , . . . , an , z) → y=z for all a1 , . . . , an ∈ N.
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4. GÖDEL’S THEOREMS
Note that in case T ⊢ b 6= c for b < c condition (4.2) follows from (4.1)
and (4.3).
Lemma. If the characteristic function cR of a relation R ⊆ Nn is representable in T , then so is the relation R itself.
Proof. For simplicity assume n = 1. Let A(x, y) be a formula representing cR . We show that A(x, 1) represents the relation R. Assume a ∈ R.
Then cR (a) = 1, hence (a, 1) ∈ GcR , hence T ⊢ A(a, 1). Conversely, assume
a∈
/ R. Then cR (a) = 0, hence (a, 1) ∈
/ GcR , hence T ⊢ ¬A(a, 1).
4.2.2. Undefinability of the notion of truth in formal theories.
Lemma (Fixed Point Lemma). Assume that all elementary functions
are representable in T . Then for every formula B(z) we can find a closed
formula A such that
T ⊢ A ↔ B(pAq).
Proof. The proof is very similar to the proof of the Semantical Fixed
Point Lemma. Let s be the elementary function from 4.1.3 and As (x1 , x2 , x3 )
a formula representing s in T . Let
˜ x (B(x) ∧ As (z, z, x)), A := C(pCq),
C := ∃
and therefore
˜ x (B(x) ∧ As (pCq, pCq, x)).
A=∃
Because of s(pCq, pCq) = pC(pCq)q = pAq we can prove in T
As (pCq, pCq, x) ↔ x = pAq,
hence by definition of A also
˜ x (B(x) ∧ x = pAq)
A↔∃
and therefore
A ↔ B(pAq).
Note that for T = Th(M) we obtain the Semantical Fixed Point Lemma
above as a special case.
Theorem. Let T be a consistent theory such that all elementary functions are representable in T . Then there cannot exist a formula B(z) defining
the notion of truth, i.e., such that for all closed formulas A
T ⊢ A ↔ B(pAq).
Proof. Assume we would have such a B(z). Consider the formula
¬B(z) and choose by the Fixed Point Lemma a closed formula A such that
T ⊢ A ↔ ¬B(pAq).
For this A we obtain T ⊢ A ↔ ¬A, contradicting the consistency of T .
4.3. UNDECIDABILITY AND INCOMPLETENESS
79
With T := Th(M) Tarski’s Undefinability Theorem is a special case.
4.3. Undecidability and Incompleteness
Consider a consistent formal theory T with the property that all recursive functions are representable in T . This is a very weak assumption, as we
shall show in the next section: it is always satisfied if the theory allows to
develop a certain minimum of arithmetic. We shall show that such a theory
necessarily is undecidable. Moreover we shall prove Gödel’s First Incompleteness Theorem, which says that every axiomatized such theory must be
incomplete. We will also prove a sharpened form of this theorem due to
Rosser, which explicitly provides a closed formula A such that neither A nor
¬A is provable in the theory T .
In this section let L again be an elementarily presented language with
0, S, = in L and T a theory containing the equality axioms EqL .
4.3.1. Undecidability.
Theorem (Undecidability). Assume that T is a consistent theory such
that all recursive functions are representable in T . Then T is not recursive.
Proof. Assume that T is recursive. By assumption there exists a formula B(z) in representing pT q in T . Choose by the Fixed Point Lemma a
closed formula A such that
T ⊢ A ↔ ¬B(pAq).
We shall prove (∗) T 6⊢ A and (∗∗) T ⊢ A; this is the desired contradiction.
Ad (∗). Assume T ⊢ A. Then A ∈ T , hence pAq ∈ pT q, hence T ⊢
B(pAq) (because B(z) represents in T the set pT q). By the choice of A it
follows that T ⊢ ¬A, which contradicts the consistency of T .
Ad (∗∗). By (∗) we know T 6⊢ A. Therefore A ∈
/ T , hence pAq ∈
/ pT q
and therefore T ⊢ ¬B(pAq). By the choice of A it follows that T ⊢ A. 4.3.2. Incompleteness.
Theorem (Gödel’s First Incompleteness Theorem). Assume that T is
an axiomatized consistent theory with the property that all recursive functions are representable in T . Then T is incomplete.
Proof. This is an immediate consequence of the fact that every axiomatized consistent theory which is complete is also recursive (a corollary in
4.1.4), and the Undecidability Theorem above.
As already mentioned, we now sharpen the Incompleteness Theorem in
the sense that we actually produce a formula A such that neither A nor ¬A
is provable. The original idea for this sharpening is due to Rosser.
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4. GÖDEL’S THEOREMS
Theorem (Gödel-Rosser). Let T be axiomatized and consistent. Assume that there is a formula L(x, y) – written x < y – such that
˜ ... ∨
˜ x = n − 1),
(4.4)
T ⊢ ∀x<n (x = 0 ∨
˜ ... ∨
˜ x=n∨
˜ n < x).
T ⊢ ∀x (x = 0 ∨
(4.5)
Assume also that every elementary function is representable in T . Then we
can find a closed formula A such that neither A nor ¬A is provable in T .
Proof. We first define RefutT ⊆ N × N by
RefutT (d, a) := Prf T (d, ¬a).
˙
Then RefutT is elementary and RefutT (d, a) if and only if d is the GN
of a derivation of the negation of a formula with GN a from a context
composed of equality axioms and formulas from AxT . Let BPrf T (x1 , x2 )
and BRefutT (x1 , x2 ) be formulas representing Prf T and RefutT , respectively.
Choose by the Fixed Point Lemma a closed formula A such that
˜y<x BRefut (y, pAq)).
T ⊢ A ↔ ∀x (BPrf (x, pAq) → ∃
T
T
A expresses its own underivability, in the form (due to Rosser): “For every
proof of me there is a shorter proof of my negation”.
We shall show (∗) T 6⊢ A and (∗∗) T 6⊢ ¬A. Ad (∗). Assume T ⊢ A.
Choose n such that
Prf T (n, pAq).
Then we also have
not RefutT (m, pAq)
for all m,
since T is consistent. Hence we have
T ⊢ BPrf T (n, pAq),
T ⊢ ¬BRefutT (m, pAq)
By (4.4) we can conclude
T ⊢ BPrf T (n, pAq) ∧ ∀y<n ¬BRefutT (y, pAq).
Hence we have
˜ x (BPrf (x, pAq) ∧ ∀y<x ¬BRefut (y, pAq)),
T ⊢∃
T
T
T ⊢ ¬A.
This contradicts the assumed consistency of T .
Ad (∗∗). Assume T ⊢ ¬A. Choose n such that
RefutT (n, pAq).
for all m.
4.4. REPRESENTABILITY
81
Then we also have
not Prf T (m, pAq)
for all m,
since T is consistent. Hence we have
T ⊢ BRefutT (n, pAq),
T ⊢ ¬BPrf T (m, pAq)
for all m.
This implies
˜y<x BRefut (y, pAq)),
T ⊢ ∀x (BPrf T (x, pAq) → ∃
T
as can be seen easily by cases on x, using (4.5). Hence T ⊢ A. But this
again contradicts the assumed consistency of T .
Finally we formulate a variant of this theorem which does not assume
that the theory T talks about numbers only. Call T a theory with defined
natural numbers if there is a formula N (x) – written N x – such that T ⊢ N 0
and T ⊢ ∀x∈N N (Sx) where ∀x∈N A is short for ∀x (N x → A). Representing
a function in such a theory of course means that the free variables in (4.3)
are relativized to N :
T ⊢ ∀y,z∈N (A(a1 , . . . , an , y) ∧ A(a1 , . . . , an , z) → y=z) for all a1 , . . . , an ∈ N.
Theorem (Gödel-Rosser). Assume that T is an axiomatized consistent
theory with defined natural numbers, and that there is a formula L(x, y) –
written x < y – such that
˜ ... ∨
˜ x = n − 1),
T ⊢ ∀x∈N (x < n → x = 0 ∨
˜ ... ∨
˜ x=n∨
˜ n < x).
T ⊢ ∀x∈N (x = 0 ∨
Assume also that every elementary function is representable in T . Then one
can find a closed formula A such that neither A nor ¬A is provable in T .
Proof. As for the Gödel-Rosser Theorem above; just relativize all quantifiers to N .
4.4. Representability
We show in this section that already very simple theories have the property that all recursive functions are representable in them.
82
4. GÖDEL’S THEOREMS
4.4.1. Weak arithmetical theories.
Theorem (Weak arithmetical theories). Let L be an elementarily presented language with 0, S, = in L and T a consistent theory with defined
natural numbers containing the equality axioms EqL and the stablity axiom
∀x,y∈N (¬¬x = y → x = y). Assume that there is a formula L(x, y) – written
x < y – such that
(4.6)
T ⊢ Sa 6= 0
for all a ∈ N,
(4.7)
T ⊢ Sa = Sb → a = b
for all a, b ∈ N,
(4.8)
the functions + and · are representable in T ,
(4.9)
T ⊢ ∀x∈N (x 6< 0),
(4.10)
˜ x = b)
T ⊢ ∀x∈N (x < Sb → x < b ∨
˜ x=b∨
˜ b < x)
T ⊢ ∀x∈N (x < b ∨
(4.11)
for all b ∈ N,
for all b ∈ N.
Then T fulfills the assumptions of the Gödel-Rosser Theorem relativized to
N , i.e.,
(4.12)
(4.13)
˜ ... ∨
˜ x = a − 1)
T ⊢ ∀x∈N (x < a → x = 0 ∨
˜ ... ∨
˜ x=a∨
˜ a < x)
T ⊢ ∀x∈N (x = 0 ∨
for all a ∈ N,
for all a ∈ N,
and every recursive function is representable in T .
Proof. (4.12) can be proved easily by induction on a. The base case
follows from (4.9), and the step from the induction hypothesis and (4.10).
(4.13) immediately follows from the trichotomy law (4.11), using (4.12).
For the representability of recursive functions, first note that the formulas x = y and x < y actually do represent in T the equality and the less-than
relations, respectively. From (4.6) and (4.7) we can see immediately that
T ⊢ a 6= b when a 6= b. Assume a 6< b. We show T ⊢ a 6< b by induction
on b. T ⊢ a 6< 0 follows from (4.9). In the step we have a 6< b + 1, hence
a 6< b and a 6= b, hence by induction hypothesis and the representability
(above) of the equality relation, T ⊢ a 6< b and T ⊢ a 6= b, hence by (4.10)
T ⊢ a 6< Sb. Now assume a < b. Then T ⊢ a 6= b and T ⊢ b 6< a, hence by
(4.11) T ⊢ a < b.
We now show by induction on the definition of µ-recursive functions,
that every recursive function is representable in T . Recall (from 4.2.1) that
the second condition (4.2) in the definition of representability of a function
automatically follows from the other two (and hence need not be checked
further). This is because T ⊢ a 6= b for a 6= b.
The initial functions constant 0, successor and projection (onto the ith coordinate) are trivially represented by the formulas 0 = y, Sx = y
and xi = y respectively. Addition and multiplication are represented in
4.4. REPRESENTABILITY
83
T by assumption. Recall that the one remaining initial function of µ· , but this is definable from the characteristic function
recursiveness is −
· b = µi (b + i ≥ a) = µi (c< (b + i, a) = 0). We now show that the
of < by a −
characteristic function of < is representable in T . (It will then follow that
· is representable, once we have shown that the representable functions are
−
closed under µ.) So define
˜ (x1 6< x2 ∧ y = 0).
A := (x1 < x2 ∧ y = 1) ∨
Assume a1 < a2 . Then T ⊢ a1 < a2 , hence T ⊢ A(a1 , a2 , 1). Now assume
a1 6< a2 . Then T ⊢ a1 6< a2 , hence T ⊢ A(a1 , a2 , 0). Furthermore notice that
∀y,z∈N (A(a1 , a2 , y) ∧ A(a1 , a2 , z) → y = z) already follows logically from the
equality axioms (by cases on a1 < a2 , using stablity of equality).
For the composition case, suppose f is defined from h, g1 , . . . , gm by
f (~a ) = h(g1 (~a ), . . . , gm (~a )).
By induction hypothesis we already have representing formulas Agi (~x, yi )
and Ah (~y , z). As representing formula for f we take
˜y~∈N (Ag (~x, y1 ) ∧ · · · ∧ Ag (~x, ym ) ∧ Ah (~y , z)).
Af := ∃
1
m
Assume f (~a ) = c. Then there are b1 , . . . , bm such that T ⊢ Agi (~a, bi ) for each
i, and T ⊢ Ah (~b, c) so by logic T ⊢ Af (~a, c). It remains to show uniqueness
T ⊢ ∀z1 ,z2 ∈N (Af (~a, z1 )∧ Af (~a, z2 ) → z1 = z2 ). But this follows by logic from
the induction hypothesis for gi , which gives
T ⊢ ∀y1i ,y2i ∈N (Agi (~a, y1i ) ∧ Agi (~a, y2i ) → y1i = y2i = gi (~a ))
and the induction hypothesis for h, which gives
T ⊢ ∀z1 ,z2 ∈N (Ah (~b, z1 ) ∧ Ah (~b, z2 ) → z1 = z2 ) with bi = gi (~a ).
For the µ case, suppose f is defined from g (taken here to be binary for
˜i (g(i, a) =
notational convenience) by f (a) = µi (g(i, a) = 0), assuming ∀a ∃
0). By induction hypothesis we have a formula Ag (y, x, z) representing g.
In this case we represent f by the formula
˜u∈N ;u6=0 Ag (v, x, u)).
Af (x, y) := N y ∧ Ag (y, x, 0) ∧ ∀v∈N (v < y → ∃
We first show the representability condition (4.1), that is T ⊢ Af (a, b) when
f (a) = b. Because of the form of Af this follows from the assumed repre˜ ... ∨
˜ v = b − 1).
sentability of g together with T ⊢ ∀v∈N (v < b → v = 0 ∨
We now tackle the uniqueness condition (4.3). Given a, let b := f (a)
(thus g(b, a) = 0 and b is the least such). It suffices to show
T ⊢ ∀y∈N (Af (a, y) → y = b).
We prove T ⊢ ∀y∈N (y < b → ¬Af (a, y)) and T ⊢ ∀y∈N (b < y → ¬Af (a, y)),
and then appeal to the trichotomy law and stablity of equality.
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4. GÖDEL’S THEOREMS
We first show T ⊢ ∀y∈N (y < b → ¬Af (a, y)). Now since, for any i < b,
T ⊢ ¬Ag (i, a, 0) by the assumed representability of g, we obtain immediately
˜ ... ∨
˜ y = b − 1)
T ⊢ ¬Af (a, i). Hence because of T ⊢ ∀y∈N (y < b → y = 0 ∨
the claim follows.
Secondly, T ⊢ ∀y∈N (b < y → ¬Af (a, y)) follows almost immediately
˜u∈N ;u6=0 Ag (b, a, u)) and the uniqueness
from T ⊢ ∀y∈N (b < y → Af (a, y) → ∃
for g, T ⊢ ∀u∈N ;u6=0 (Ag (b, a, u) → u = 0). This completes the proof.
4.4.2. Robinson’s theory Q. We conclude this section by considering
a special and particularly simple arithmetical theory due originally to Robinson. Let L1 be the language given by 0, S, +, · and =, and let Q be the theory
determined by the axioms EqL1 , stability of equality ¬¬x = y → x = y and
(4.14)
Sx 6= 0,
(4.15)
Sx = Sy → x = y,
(4.16)
x + 0 = x,
(4.17)
x + Sy = S(x + y),
(4.18)
x · 0 = 0,
(4.19)
x · Sy = x · y + x,
˜z (x + Sz = y) ∨
˜z (y + Sz = x).
˜ x=y∨
˜∃
∃
(4.20)
Theorem (Robinson’s Q). Every consistent theory T ⊇ Q fulfills the
assumptions of the Gödel-Rosser Theorem w.r.t. the definition L(x, y) :=
˜ z (x + Sz = y) of the <-relation. In particular, every recursive function is
∃
representable in T .
Proof. We show that T satisfies the conditions of the previous theorem.
For (4.6) and (4.7) this is clear. For (4.8) we can take x + y = z and x · y = z
˜z (x + Sz = 0); this
as representing formulas. For (4.9) we have to show ¬∃
follows from (4.17) and (4.14). For the proof of (4.10) we need the auxiliary
proposition
(4.21)
˜y (x = 0 + Sy),
˜∃
x=0∨
which will be attended to below. Assume x + Sz = Sb, hence also S(x + z) =
˜ y′ (x + Sy ′ = b) ∨
˜ x = b. But
Sb and therefore x + z = b. We must show ∃
this follows from (4.21) for z. In case z = 0 we obtain x = b, and in case
˜ y (z = 0 + Sy) we have ∃
˜ y′ (x + Sy ′ = b), since 0 + Sy = S(0 + y). Thus
∃
(4.10) is proved. (4.11) follows immediately from (4.20). For the proof of
(4.21) we use (4.20) with y = 0. It clearly suffices to exclude the first case
˜ z (x + Sz = 0). But this means S(x + z) = 0, contradicting (4.14).
∃
4.4. REPRESENTABILITY
85
Corollary (Essential undecidability of Q). Every consistent theory
T ⊇ Q in an elementarily presented language L(T ) is non-recursive.
Proof. This follows from the theorem above and the Undecidability
Theorem in 4.3.1.
Corollary (Undecidability of logic). The set of formulas derivable in
the classical fragment of minimal logic is non-recursive.
Proof. Otherwise Q would be recursive, because a formula A is derivable in Q if and only if the implication B → A is derivable, where B is the
conjunction of the finitely many axioms and equality axioms of Q.
Remark. Note that it suffices that the underlying language contains
one binary relation symbol (for =), one constant symbol (for 0), one unary
function symbol (for S) and two binary functions symbols (for + and ·). The
study of decidable fragments of first order logic is one of the oldest research
areas of Mathematical Logic. For more information see Börger et al. (1997).