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The Completeness Theorem of Godel 2 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 Soundness Theorem and Validity 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. 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 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. -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. 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; --------~-------- 62 RESONANCE I August 2001 GENERAL I ARTICLE (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). -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 ,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.
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