In mathematical logic, Craig's theorem states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem, although both results are named after the same mathematician, William Craig.
Let be an enumeration of the axioms of a recursively enumerable set T of first-order formulas. Construct another set T* consisting of
for each positive integer i. The deductive closures 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 or of the form
Since each formula has finite length, it is checkable whether or not it is or of the said form. If it is of the said form and consists of j conjuncts, it is in T* if the (reoccurring) conjunct is ; otherwise it is not in T*. Again, it is checkable whether the conjunct is in fact by going through the enumeration of the axioms of T and then checking symbol-for-symbol whether the expressions are identical.
Primitive recursive axiomatizations
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 with
one instead replaces it with
where f(x) is a function that, given i, returns a computation history showing that 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 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.
- William Craig. On Axiomatizability Within a System, The Journal of Symbolic Logic, Vol. 18, No. 1 (1953), pp. 30-32.