GENERAL
I
ARTICLE
The Completeness Theorem of Godel
2. Henkin's Proof for First Order Logic
S M Srivastava
In Part 11 of the article, we introduced the basic
notions and techniques of mathematical logic. In
this part, we present the completeness theorem
of first order logic proved first by Godel in 1929.
We give a sketch of the proof due to Henkin.
S M Srivastava is with the
Indian Statistical,
Institute, Calcutta. He
received his PhD from
the Indian Statistical
Institute in 1980. His
research interests are in
descriptive set theory.
I Part 1. An Introduction to Mathematical Logie, Resom;mce,
Vo1.6, No.7, pp.29-41, 2001.
Soundness Theorem and Validity Theorem
As before, the logical symbols ...." V, 1\, -+, 3, V denote
"not", "or", "and", "implies", "for some", "for all" , respectively.
We start with the syntactical definition of a proof. This
is based on two very important observations. We know
that some formulae are true in a structure because of
particular properties of the structure. On the other
hand, some formulae, e.g., those of the form ..,A V A or
x = x or Ax[t] -+ 3xA, are true in all structures simply
because of the meaning of the logical symbols. Similarly
sometimes a formula is inferred from a set of formulae
because of the meaning of logical symbols. For example,
the formula B is true in all those structures in which A
and A -+ B are true and 3xA -+ B is true in all those
structures in which A -+ B is true provided x is not
free in B. To define a proof syntactically, we fix some
formulae which are true in all structures and call them
logical axioms. Further, we fix some rules of inference.
For simplicity, we take up the case of propositional logic
first.
1a. Syntax of Propositional Logic
Any formula of the form ..,A V A is called a propositional
axiom. These are all the logical axioms of the proposi-
-60---------------------------------------------------------~---------------------------------RESONANCE I August 2001
GENERAL I ARTICLE
tionallogic. Rules of inference of the propositional logic
are
The definition of
proof is such that
once a sequence
Expansion Rule. Infer B V A from A.
is given, it can be
mechanically
Contraction Rule. Infer A from A V A.
Associative Rule. Infer (A V B) V C from A V (B V C).
verified whether it
is a proof or not.
Cut Rule. Infer B V C from A V Band -,A V C.
Note that the conclusion of any rule of inference is true
in any structure in which its hypothesis is true.
Definition. Let A be a set of formulae not containing
any logical axiom. (For reasons that will become clear
below, elements of A will be called non-logical axioms.)
A proof in A is a finite sequence of formulae AI, A 2 ,
An such that each Ai is either a logical axiom or a
non-logical axiom or can be inferred from formulae A j ,
j < i, using one of the rules of inference. In this case
we call the above sequence a proof of An in A. It is
worth noting that the definition of proof is such that
once a sequence is given, it can be mechanically verified
whether it is a proof or not. If A has a proof in A, we
say that A is a theorem of A and write A" ~ A. For
,An ~ A instead of
brevity, we shall write AI, A 2,
{AI, A2,
,An} ~ A and ~ A instead of 0 ~ A.
• ".1 . ,
Observe that if there is a sequence AI, A2, ., An such
that each Ai is either a theorem of A or can be inferred
from formulae A j, j < i, using one of the rules of inference, then An is a theorem of A.
Proposition 1.1. A V B
~
B
V
A.
Proof. Consider the sequence
A VB,
-,A V A, B V A.
-RE-S-O-N-A-N-C-E-I-A-U-9-U-st--2-0-01-----------~-----------------------------61
GENERAL
I
ARTICLE
The first element of it is a non-logical axiom, the second
one a logical one and the third one follows from the first
two by the cut rule.
Proposition 1.2. (Detachment rule)
A, A ~ B ~ B.
Proof. First note that, by the expansion rule, B V A
is a theorem of any A containing A. Hence, by the last
proposition, A V B is a theorem of any A containing A.
Now consider the sequence
AVB,
A~B,
BVB, B.
The first formula of this sequence is shown above to be
a theorem of {A, A ~ B}; the second formula is a nonlogical axiom; the third formula is inferred from the first
two by the cut rule (recall that A ~ B is an abbreviation
for -,A VB); the last formula is inferred from the third
formula by the contraction rule.
Now, by induction on n, we easily get
Corollary 1.3. Al,
,An,A 1
~
·A n
~
B
~
B.
The following important result is quite easy to prove.
Theorem 1.4. (Soundness theorem) For any formula
A and any set of formulae A, A ~ A implies A F A.
Proof. Let Al,. ,An be a proof of A in A. By induction, we can easily show that for each i, 1 :::; i :::; n,
A pAi.
1 h. Syntax of First Order Logic
In this section we shall extend the notion of the proof
to first order logic. Besides propositional axioms (Le.,
formulae of the form -,A V A), other logical axioms of L
are:
(a) identity axioms: these are formulae of the form
x = x, where x is a variable;
--------~-------RESONANCE I August 2001
62
GENERAL I ARTICLE
(b) equality axioms: these are formulae of the form
---?
Xn
= Yn
---?
f Xl
. Xn
=
fYl
. Yn
Every logical
axiom and every
rule of inference of
propositional logic
or formulae of the form
is also so in first
order theories.
(c) substitution axioms: these are formulae of the
form Ax[t] ---? 3xA, where t is any term substitutable for X in A;
Besides expansion, contraction, associative and cut rules,
first order logic has one more rule of inference:
3-introduction rule: If x is not free in B, infer 3xA
B from A ---? B.
---?
We again note that each logical axiom is true in every
structure of L. Also, the conclusion of each rule of inference is true in any structure in which its hypothesis
is true.
A first order theory T consists of a first order language
L = L(T), logical axioms and logical rules of inference
of L and a set of formulae of L (other tha:n the logical
axioms of L) called the non-logical axioms of T A proof
in T, a theorem in T etc. are defined exactly as before.
We shall write T I- A or simply I- A (when T is understood) to say that A is a theorem of T A model of T is
a structure M of L(T) in which all non-logical axioms
of T are true. We say that a formula A of L(T) is valid
in T if it is true in every model of T In this case we
write T F A or simply F A.
In the rest of this section, T denotes a first order theory
with language L and by a formula, we mean a formula
of L.
We prove the following theorem in exactly the same way
as we, proved the soundness theorem (1.4).
-RE-S-O-N-A-N-C-E-I--AU-9-U-st--2-0-01----------.-~----------------------------6-3
GENERAL
I
ARTICLE
Theorem 1.5. (Validity theorem) T
~
A implies T
F
A.
Since every logical axiom and every rule of inference
of propositional logic are also those of first order logic,
results such as 1.1-1.3 hold for first order theories also.
We shall need many such proof-theoretic results for first
order theories. For instance, later we shall use
Proposition 1.6. If all
free terms, then
,bn are variable-
and
However, we shall omit this tedious and somewhat dull
part of mathematical logic. We shall only state such
results whenever needed. The interested reader may see
Shoenfield's book[1].
We now introduce a few more syntactical notions. A
first order language L' is called an extension of the first
order language L, if every constant symbol of L is a
constant symbol of L' every n-ary function symbol of L
is a n-ary function symbol of L' and every n-ary relation
symbol of L is a n-ary relation symbol of L'. A theory T'
is called an extension of the first order theory T if L(T')
is an extension of L(T) and every non-logical axiom of
T is a theorem of T'.
Exercise 1.7. Show that if T' is an extension of T then
every theorem of T is a theorem of T'
Let T' be an extension of T The theory T' is called a
simple extension of T if L(T') = L(T). If r is a set of
formulae T T [r) will denot~ the simple extension of T
whose non-logical axioms are the non-logical axioms of
T together with the formulae A E r The theory T' is
--------~-------64
RESONANCE I August 2001
GENERAL I ARTICLE
called a conservative extension of T if every formula of
L(T) that is a theorem of T' is a theorem of T
We call a theory T inconsistent if for some formula A,
T f- A as well as T f- ..,A. We call theory T consistent
if it is not inconsistent. We call a formula A decidable
in T, if either A or ..,A is a theorem of T A theory T
is called complete if it is consistent and if ev~ry closed
form ula is decidable in T
Exercise 1.8. Show that T is inconsistent if and only
if every formula A is a theorem of T
Exercise 1".9. Show that every conservative extension
of a consistent theory is consistent.
We shall omit the proof of the next result.
Proposition 1.10. If A is a closed formula that is not
decidable in T, then both T[A] and T[..,A] are consistent.
We can restate the validity theorem in a very interesting
way.
Theorem 1.11. Every theory having a model is consistent.
We close this section by proving rather a useful result.
Theorem 1.12. Every consistent theory admits a simple complete extension.
Proof. For simplicity we present the proof under the
additional assumption that T has only countably many
non-logical symbols. (There are well-known techniques
such as Zorn's lemma or transfinite induction to generalize our argument for all T quite easily.) Then, there
enuare only countably many formulae. Let AI, A2,
merate all the closed formulae of T Inductively we deas follows: Let B 1 be the first
fine formulae B 1, B2,
Ai that is not decidable in T Having defined BI, B 2 ,
., B n , let Bn+l be the first Ai that is not decidable in
-RE-S-O-N-A-N-C-E-I-A-U-9-US-t--2-00-1-----------~-----------------------------65
GENERAL
I
ARTICLE
Deductions as
Bn] is consistent.
done by
Set r = {Bn : n ~ I}. Clearly T[r] is a simple extension
of TIt is consistent because any proof in T [r], being a
sequence of finite length, is a proof in T[Bl'
, Bn] for
some n. So, if both A and -.A are theorems of T[r], they
are both. theorems of T[Bl,
,Bn] for some n. This is
, Bn] is consistent.
impossible because T[BI,
mathematicians is
a purely
mechanical
process.
2. Completeness Theorems
We have now defined a proof both semantically and syntactically. A natural and important question arises. Are
the two definitions of a proof equivalent? The answer
is yes. Thus, deductions as done by mathematicians is
a purely mechanical process. This is probably the first
non-trivial theorem of logic, called the completeness theorem. Quite naturally, the discovery of the notion of a
proof turned out to be of fundamental importance for
independence proofs, in particular, and for the foundations of mathematics, in general. For propositional logic,
the completeness theorem was independently proved by
Emil Post in 1921 and Paul Bernays in 1926. For first
order logic having only countably many non-logical symbols, it was first proved by Godel in 1929.
The main aim of this article is to present this result.
However, it is not possible to give here a complete proof.
We shall only give a sketch of the proof. The proof of
the completeness theorem for first order logic presented
here is due to Henkin (1949).
2a. Completeness Theorem for Propositional Logic
Throughout this section, L denotes a language for propositional logic. Formulae, proofs, etc. are those in L.
Theorem 2.1. (Tautology theorem) If A is a tautological consequence of AI, A2, ., An, then
,An I- A.
----~--~-------66
RESONANCE I August 2001
GENERAL
I
ARTICLE
Proof. Since A is a tautological consequence of At, A 2 ,
., An, Al ~
~ An ~ A is a tautology.
Claim. Every tautology A is a theorem.
Quite naturally, the
discovery of the
notion of a proof
turned out to be of
~ An ~ A.
Assuming the claim, we have /- Al ~
The result now easily follows from the corollary 1.3 to
the detachment rule.
fundamental
To prove our claim, we shall prove that
proofs, in
importance for
independence
particular, and for
(*) whenever Al V A2 V
it is a theorem.
V An (n ~ 2) is a tautology,
Assuming (*), the claim can be proved as follows. Since
A is a tautology, so is A V A. So, by (*), /- A V A. The
claim now follows from the contraction rule.
the foundations of
mathematics, in
general.
We shall prove (*) by induction on the sum of lengths of
Ai'S. Suppose each Ai is either an atomic formula or the
negation of an atomic formula. Since Al V A2 V . V An
is a tautology, there exist 1 ~ i i= j ~ n such that Aj is
,Ai. Hence /- Aj V Ai because Aj V Ai is a propositional
axiom. The result in this case will follow from
(I) If k ~ 1, l ~ 1, and il, i2,
., il are among 1, 2,
., k, thenA i1 V Ai2 V
V Ail/- Al V A2 V
V Ak.
We now assume that some Ai is neither an atomic formula nor the negation of an atomic formula. By (I),
without any loss of generality, we assume that Al has
this property.
V An is a
Suppose Al is B V C. Since Al V A2 V
V An. But the sum
tautology, so is B V C V A2 V
of the lengths of B, C, A 2 ; ., An is less than those
of At, A 2, ., An. Hence, by the induction hypothesis,
/- B V C V A2 V
V An. Hence, /- Al V A2 V
V An by
the associative rule.
Suppose Al is "B. Since Al V A2 V· . V An is a tautology, B V A2 V· . V An is a tautology. Hence, by induction
--------~-------RESONANCE I August 2001
67
GENERAL I ARTICLE
hypothesis, it is a theorem. The result in this case will
follow from
(II) P V Q r- "P V Q.
Finally, suppose Al is ,(B V C). Since Al V A2 V· . V An
is a tautology, ,BvA2V' ·VA n and ,CVA2V' ·VA n are
tautologies. Hence, they are theorems by the induction
hypothesis. The result in this case will follow from
(III) ,P V R, ,Q V R
r- ,(P V Q) V R.
Proofs of (I), (II) and (III) are omitted.
Our next result is an extension of the tautology theorem
(2.1) to the first order theories. To do this we note
that (I), (II), (III) as stated in the proof of 2.1 hold
for first order theories also. Therefore, if we replace the
term 'atomic formulae' by 'elementary formulae' in the
proof of the tautology theorem, we get the following two
equivalent results.
Theorem 2.2. (Tautology theorem) Let T be any first
order theory. If T r- AI,
., T r- An and if A is a
tautological consequence of AI, ., An, then T r- A.
Theorem 2.3. Every tautology in a first order theory
T is a theorem of T
2b. Completeness Theorem for First Order Logic
In this section we give a sketch of the proof of the completeness theorem for first order logic.
Theorem 2.4. (Completeness theorem) Every consistent first order theory T has a model.
Since we have o~ly syntactical objects at hand, a model
of T has to be built out of these. Since syntactical objects that designate individuals of a model are variablefree terms of the language of T, it seems quite natural
to start with these.
-68---------------------------~----------------------------RESONANCE I August 2001
GENERAL
I
ARTICLE
Let N be the set of all variable-free terms. It is quite
possible that theorems of T may force two variable-free
terms to designate the same individuals. Therefore, we
define a binary relation on N as follows:
a rv b if T I- a
= b,
where a, b belong to N It can be shown that rv is an
equivalence relation on N We set M to be the set of
rv-equivalence classes. For any a E N, raj will denote
the equivalence class containing a. The set M will be
the universe of our intended model M.
Can M be empty? It is not so, if there is at least one
constant symbol. To start with we assume that T has
at least one constant symbol.
We now define the interpretations of the non-logical
symbols of T in M in a natural way.
M(c)
= [c],
and
[anD if and only if T I- pal
an.
In the above definitions, c is a constant symbol (so a
,an are variable-free terms, f
variable-free term), at,
a n-ary function symbol and p a n-ary relation symbol.
The above functions and relations are well-defined by
1.6. We have now defined a structure for the language
of T This structure is called the canonical structure of
T Is the canonical structure of T a model of T? We
investigate this question now.
Call T a Henkin theory if for every closed formula of the
form :3xA there is a constant symbol, say c, in L(T) such
that T I- :3xA ---+ Ax[cJ. In particular, M is non-empty
if T is Henkin.
-RE-S-O-N-A-N-C-E-I-A-U-9-US-t--20-0-1-----------~-----------------------------'9
GENERAL
I ARTICLE
Every theory T has a Henkin extension T' which is min.,.
imal in some sense. To see this set, To = T Let Tn be
defined. We define a theory Tn+l as follows: for each
closed formula of the form 3xA of Tn, add a new sym.,.
bol, say C3xA, and a new axiom 3xA ---+ Ax [C3xA]. Now
let T' be the theory, whose language is the 'union' of the
languages of Tn's and whose a..xioms are axioms of Tn's.
Clearly, T' is a Henkin extension of T
The following is somewhat a deep fact.
Theorem 2.5 The Henkin extension T' of T is a con.,.
servative extension of T In particular, T' is consistent
if T is so.
We shall omit its proof.
Claim. If T is a complete Henkin theory, then the
canonical structure for T is a model of T
We show this by showing the following. For every closed
formula A,
T
r- A if and only if M 1= A.
(It can be justified that it is sufficient to consider the
closed formulae only.) The proof of (*) proceeds by in.,.
duction on the number of times the logical symbols V,
-, and 3 occur in A. This number is called the height of
A. By definition of M, (*) holds for all atomic A.
Suppose B is the formula -,A and that (*) holds for A.
Let T r- B. Since T is consistent, it follows that T If A.
By our assumptions on A, it follows that M ~ A. But
thenM 1= B. Conversely, suppose T If B. Since T is
complete, this implies that T I- A. By the induction
hypothesis on A, M 1= A. So, M ~ B.
Now suppose A = BVC, and (*) holds for Band C. We
can show that (*) holds for A using similar arguments
and only using the assumption that T is complete.
Suppose B is the formula 3xA and (*) holds for all for.,.
--------~-------I August 2001
70
RESONANCE
GENERAL
I
ARTICLE
mulae of heights less than the height of B. Let T r 3xA.
Since T is a Henkin theory, there is a constant symbol
e such that :3xA ~ Ax[e] is a theorem of T By the
detachment rule, it follows that T r Ax[e]. By the induction hypothesis, M ~ Ax[e], and hence M 1= B.
Conversely, suppose M F B. Then there is a m E M
such that M ~ AxIi m ]. Let m = [a] for some variablefree term a. So, M FAx[a]. By induction hypothesis,
T r A x [a1. Since Ax[a1 ~ :3xA is substitution axiom,
the detachment rule gives us T r B showing that every
Henkin theory has a model.
a
It is quite easy to complete the proof of the completeness
theorem now. Let T be a consistent theory. By 2.5, we
get a conservative extension T' of T which is Henkin.
Since T is consistent, by 1.9, so is T'. By 1.12, T' has a
simple complete extension Til Clearly Til is a complete
Henkin theory. As shown, there is a model M of Til.
Evidently, M is a model of T
Suggested Reading
[1] Joseph R
Shoenfield,M~Logic,
~
,
I'
Address for Correspondence
S M Srivastava
Stat-Math Unit
Indian Statistical Institute
203 BT Road
Calcutta 700 035, India .
e-mail: [email protected]
Addison-Welley, 1967.
Kurtesy Godel
The completeness theorem of Kurt Godel
shows consistent theory has a model
(pronounce the last word - if you are able
- the rhyme would still be reasonable) Let's sing its praise, let's all yodel!
If you think this ends it, wait!
We still need to decide the fate
of statements not denied right out;
indeed, are true without a doubt
but impossible to demonstrate!
.. Kanakku Puly ..
-RE-S-O-N-A-N-C-E-I--AU-9-U-st--2-0-01-----------~------------------------------n