Craig's theorem

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. Distinguish this result from the more well-known Craig interpolation theorem.

Proof. Let $A_{1},A_{2},\dots$ be an enumeration of the axioms of a recursively enumerable set of first-order formulas. Construct another set T* consisting of

$\underbrace {A_{i}\land \dots \land A_{i}} _{i}$

for each positive integer i. Clearly the deductive closures of T* and T are equivalent. A decision procedure for T* lends itself according to the following informal reasoning. Each member of T* is either $A_{1}$ or of the form

$\underbrace {B_{j}\land \dots \land B_{j}} _{j}$.

Since each formula has finite length, it is checkable whether or not it is $A_{1}$ 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 $A_{j}$; otherwise it is not in T*. Again, it is checkable whether it is in fact $A_{n}$ by going through the enumeration of the axioms of T and then checking symbol-for-symbol whether the expressions are identical.

References

William Crag. On Axiomatizability Within a System, The Journal of Symbolic Logic, Vol. 18, No. 1 (1953), pp. 30-32.