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chapter14(1hcompl)

# chapter14(1hcompl) - 1 Completeness Theorem for First Order...

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Unformatted text preview: 1 Completeness Theorem for First Order Logic There are many proofs of the Completeness Theorem for First Order Logic. We follow here a version of Henkin’s proof, as presented in the Handbook of Mathe- matical Logic . It contains a method for reducing certain problems of first-order logic back to problems about propositional logic. We give independent proof of Compactness Theorem for propositional logic. The Compactness Theorem for first-order logic and L¨owenheim-Skolem Theorems and the G¨odel Completeness Theorem fall out of the Henkin method. 1.1 Compactness Theorem for Propositional Logic Let L = L ( P , F , C ) be a first order language with equality. We assume that the sets P, F, C are infinitely enumerable. We define a propositional logic within it as follows. Prime formulas We consider a subset P of the set F of all formulas of L . Intuitively these are formulas of L which are not direct propositional com- bination of simpler formulas, that is, atomic formulas ( A F ) and formulas beginning with quantifiers. Formally, we have that P = { A ∈ F : A ∈ A F or A = ∀ xB, A = ∃ xB for B ∈ F} . Example 1.1 The following are primitive formulas. R ( t 1 ,t 2 ) , ∀ x ( A ( x ) ⇒ ¬ A ( x )) , ( c = c ) , ∃ x ( Q ( x,y ) ∩ ∀ yA ( y )) . The following are not primitive formulas. ( R ( t 1 ,t 2 ) ⇒ ( c = c )) , ( R ( t 1 ,t 2 ) ∪ ∀ x ( A ( x ) ⇒ ¬ A ( x )) . Given a set P of primitive formulas we define in a standard way the set P F of propositional formulas as follows. Propositional formulas The smallest set P F ⊂ F such that 1. P ⊂ P F 2. If A,B ∈ P F , then ( A ⇒ B ) , ( A ∪ B ) , ( A ∩ B ) , and ¬ A ∈ P F is called a set of propositional formulas of the first order language L . We define propositional semantics for propositional formulas in P F as follows. 1 Truth assignment Let P be a set of prime formulas and { T,F } be a two element set, thought as the set of logical values ”true” and ”false”. Any function v : P-→ { T,F } is called truth assignment (or variable assignment). Let B = ( { T,F } , ⇒ , ∪ , ∩ , ¬ ) be a two-element Boolean algebra and PF = ( P F , ⇒ , ∪ , ∩ , ¬ ) a similar algebra of propositional formulas. We extend v to a homomorphism v * : PF-→ B in a usual way, i.e. we put v * ( A ) = v ( A ) for A ∈ P , and for any A,B ∈ P F , v * ( A ⇒ B ) = v * ( A ) ⇒ v * ( B ), v * ( A ∪ B ) = v * ( A ) ∪ v * ( B ), v * ( A ∩ B ) = v * ( A ) ∩ v * ( B ), v * ( ¬ A ) = ¬ v * ( A ). Propositional Model A truth assignment v is called a propositional model for a formula A ∈ P F iff v * ( A ) = T ....
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chapter14(1hcompl) - 1 Completeness Theorem for First Order...

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