Craig's observation

‘Craig's theorem’ (Craig, 1953), as philosophers call it, is actually a corollary to an observation. The observation is that (I) Every theory that admits a recursively enumerable set of axioms can be recursively axiomatized.

Some explanations are in order here: (1) A theory is an infinite set of wffs (well-formed formulas) which is closed under the usual rules of deduction. One way of giving a theory T is to specify a set of sentences S(called the axioms of T) and to define T to consist of the sentences S together with all sentences that can be derived from (one or more) sentences in S by means of logic. (2) If T is a theory with axioms S, and S' is a subset of T such that every member of S can be deduced from sentences in S', then S' is called an alternative set of axioms for T. Every theory admits of infinitely many alternative axiomatizations – including the trivial axiomatization, in which every member of T is taken as an axiom (i.e. S = T). (3) A set S is called recursive if and only if it is decidable – i.e. there exists an effective procedure for telling whether or not an arbitrary wff belongs to S. (This is not the mathematical definition of ‘recursive’, of course, but the intuitive concept which the mathematical definition captures.) For ‘effective procedure’ one can also write ‘Turing machine’ (cf. Davis, 1958.).