In mathematical logic, Craig's Theorem—not to be confused with Craig's Interpolation Theorem—states that any recursively enumerable theory is recursively axiomatizable. Its epistemological interest lies in its possible use as a method of eliminating "theoretical content" from scientific theories.

Proof of Craig's Theorem

Assume that S is a deductively closed set of sentences, the elements of which may be recursively enumerated thus F(0), F(1), …, F(n), … where F is a recursive function from natural numbers to sentences (we assume that expressions, sentences, etc., have been Gödel-coded in some manner). The set of theorems of an axiomatic theory is automatically recursively enumerable. But in general a recursively enumerable set is not automatically recursive. An example of a recursively enumerable set that is non-recursive is the set of logical truths in a first-order language with a single dyadic predicate. This follows from...