The system A′

We have seen that the system AI is incomplete and that any extension of it which remains a formal system will also be incomplete. We could add, as extra axioms, some A-true but AI-unprovable statements so as to obtain more A-true statements as theorems in the resulting extended system. But as long as we have a formal system it will still be incomplete and an irresolvable statement can be constructed on the same lines as before.

We can do this programme in a systematic manner as follows: We have an effective method for constructing an irresolvable ℒ-true ℒ-statement in a formal system ℒ which contains recursive number theory and negation. Call this ℒ-statement G{ℒ}. We first form G{AI} and then the system G′ which consists of the system AI with the extra axiom G{AI}; having formed G(λ) we construct G(Sλ) by adding the extra axiom G{G(λ)}. Having formed the systems AI, G′,…, G(λ),… we then form the system G* as the union of all the systems AI, G′,…, i.e. the system AI with all the extra axioms we added in forming the systems G′,…; this system again will be formal, hence we can form G{G*} and the system G*′ which is the system G* plus the extra axiom G{G*}. So we can proceed through the constructive ordinals. But we shall have to stop before we come to the end of the constructive ordinals, otherwise we shall cease to have a formal system.