GENERAL I ARTICLE The of Godel 2. Henkin's Proof for First Order

S M Srivastava

In Part 11 of the article, we introduced the basic notions and techniques of . 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 Theorem and Theorem Indian Statistical, As before, the logical symbols ...." V, 1\, -+, 3, V denote Institute, Calcutta. He received his PhD from "not", "or", "and", "implies", "for some", "for all" , re­ the Indian Statistical spectively. Institute in 1980. His research interests are in We start with the syntactical definition of a proof. This descriptive set theory. 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 I Part 1. An Introduction to Math­ hand, some formulae, e.g., those of the form ..,A V A or ematical Logie, Resom;mce, x = x or Ax[t] -+ 3xA, are true in all structures simply Vo1.6, No.7, pp.29-41, 2001. 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. 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-

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tionallogic. Rules of inference of the propositional logic The definition of are 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. verified whether it is a proof or not. Associative Rule. Infer (A V B) V C from A V (B V C).

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 ,

• ".1 ., 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 brevity, we shall write AI, A 2, ,An ~ A instead of {AI, A2, ,An} ~ A and ~ A instead of 0 ~ A. 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 infer­ ence, 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.

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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. N ow 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 non­ logical 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,A1 ~ ·An ~ 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 in­ duction, 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;

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(b) equality axioms: these are formulae of the form Every logical axiom and every ---? Xn = Yn ---? f Xl . Xn = fYl . Yn rule of inference of or formulae of the form propositional logic is also so in first order theories.

(c) substitution axioms: these are formulae of the

form Ax[t] ---? 3xA, where t is any term substi­ tutable 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 in­ ference 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 under­ stood) 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).

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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 ,bn are variable- free terms, then

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 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 con­ sistent. We close this section by proving rather a useful result. Theorem 1.12. Every consistent theory admits a sim­ ple 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 gener­ alize our for all T quite easily.) Then, there are only countably many formulae. Let AI, A2, enu­ merate all the closed formulae of T Inductively we de­ fine formulae B 1, B2, as follows: Let B 1 be the first 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

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Deductions as Bn] is consistent. done by Set r = {Bn : n ~ I}. Clearly T[r] is a simple extension mathematicians is of TIt is consistent because any proof in T [r], being a a purely sequence of finite length, is a proof in T[Bl' , Bn] for mechanical some n. So, if both A and -.A are of T[r], they process. are both. theorems of T[Bl, ,Bn] for some n. This is impossible because T[BI, , Bn] is consistent. 2. Completeness Theorems We have now defined a proof both semantically and syn­ tactically. 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 the­ orem. 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 founda­ tions 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 sym­ bols, 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 propo­ sitional logic. Formulae, proofs, etc. are those in L.

Theorem 2.1. ( theorem) If A is a tautolog­ ical consequence of AI, A2, ., An, then

,An I- A.

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Proof. Since A is a tautological consequence of At, A 2 , Quite naturally, the ., An, Al ~ ~ An ~ A is a tautology. discovery of the Claim. Every tautology A is a theorem. notion of a proof turned out to be of Assuming the claim, we have /- Al ~ ~ An ~ A. fundamental The result now easily follows from the corollary 1.3 to importance for the detachment rule. independence To prove our claim, we shall prove that proofs, in particular, and for (*) whenever Al V A2 V V An (n ~ 2) is a tautology, the foundations of it is a theorem. mathematics, in Assuming (*), the claim can be proved as follows. Since general. A is a tautology, so is A V A. So, by (*), /- A V A. The claim now follows from the contraction rule. 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, thenAi1 V Ai2 V V Ail/- Al V A2 V V A k . We now assume that some Ai is neither an atomic for­ mula nor the negation of an atomic formula. By (I), without any loss of generality, we assume that Al has this property.

Suppose Al is B V C. Since Al V A2 V V An is a tautology, so is B V C V A2 V V An. But the sum 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 tautol­ ogy, B V A2 V· . V An is a tautology. Hence, by induction

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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' ·VAn and ,CVA2V' ·VAn 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 com­ pleteness theorem for first order logic.

Theorem 2.4. (Completeness theorem) Every consis­ tent 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 ob­ jects that designate individuals of a model are variable­ free terms of the language of T, it seems quite natural to start with these.

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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 variable-free term), at, ,an are variable-free terms, f 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.

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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 and only using the assumption that T is complete. Suppose B is the formula 3xA and (*) holds for all for.,.

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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 in­ duction 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 variable­ free term a. So, M FAx[a]. By induction hypothesis, T r Ax[a1. Since Ax[a1 ~ :3xA is a substitution axiom, the detachment rule gives us T r B showing that every Henkin theory has a model. 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 Address for Correspondence Henkin theory. As shown, there is a model M of Til. S M Srivastava Evidently, M is a model of T Stat-Math Unit Indian Statistical Institute 203 B T Road Suggested Reading Calcutta 700 035, India. e-mail: [email protected] [1] Joseph R Shoenfield,M~Logic, Addison-Welley, 1967.

Kurtesy Godel

, I' 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 ..

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