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| In [[mathematical logic]], '''Craig's theorem''' states that any [[recursively enumerable set]] of [[well-formed formula]]s of a [[first-order language]] is (primitively) recursively axiomatizable. This result is not related to the well-known [[Craig interpolation]] theorem.
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| == Recursive axiomatization ==
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| Let <math>A_1,A_2,\dots</math> be an enumeration of the axioms of a recursively enumerable set T of first-order formulas. Construct another set T* consisting of
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| :<math>\underbrace{A_i\land\dots\land A_i}_i</math>
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| for each positive integer ''i''. The [[deductive closure]]s of T* and T are thus equivalent; the proof will show that T* is a decidable set. A decision procedure for T* lends itself according to the following informal reasoning. Each member of T* is either <math>A_1</math> or of the form
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| :<math>\underbrace{B_j\land\dots\land B_j}_j.</math> | |
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| Since each formula has finite length, it is checkable whether or not it is <math>A_1</math> or of the said form. If it is of the said form and consists of ''j'' conjuncts, it is in T* if it is the expression <math>A_j</math>; otherwise it is not in T*. Again, it is checkable whether it is in fact <math>A_n</math> by going through the enumeration of the axioms of T and then checking symbol-for-symbol whether the expressions are identical.
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| == Primitive recursive axiomatizations ==
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| The proof above shows that for each recursively enumerable set of axioms there is a recursive set of axioms with the same deductive closure. A set of axioms is [[primitive recursive]] if there is a primitive recursive function that decides membership in the set. To obtain a primitive recursive aximatization, instead of replacing a formula <math>A_i</math> with
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| :<math>\underbrace{A_i\land\dots\land A_i}_i</math>
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| one instead replaces it with
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| :<math>\underbrace{A_i\land\dots\land A_i}_{f(i)}</math> (*)
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| where ''f''(''x'') is a function that, given ''i'', returns a computation history showing that <math>A_i</math> is in the original recursively enumerable set of axioms. It is possible for a primitive recursive function to parse an expression of form (*) to obtain <math>A_i</math> and ''j''. Then, because [[Kleene's T predicate]] is primitive recursive, it is possible for a primitive recursive function to verify that ''j'' is indeed a computation history as required.
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| ==References==
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| * [[William Craig (logician)|William Craig]]. '''On Axiomatizability Within a System''', ''The Journal of Symbolic Logic'', Vol. 18, No. 1 (1953), pp. 30-32.
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| [[Category:Computability theory]]
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| [[Category:Theorems in the foundations of mathematics]]
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Nice to satisfy you, I am Marvella Shryock. For years he's been working as a receptionist. Minnesota has usually been his at home std test but his wife desires them to move. To gather coins is 1 of the issues I love most.