 So far, we focussed on the operationalist motives for reformulating QFT and some of its consequences. Now we will distinguish different, partly competing ways of implementing these general ideas. The second motive—mathematical rigour—consists foremost in the quest towards a concise axiomatic formulation, instead of the grab bag of conventional QFT, with its numerous mathematically dubious, even though successful, approximation techniques. This quest comprises three parts, namely, first, the choice of those entities upon which the axioms are to be imposed, second, the choice of appropriate axioms, and, third, the proof that one has actually achieved an axiomatic reformulation of conventional QFT, which can reproduce all the established empirical and theoretical successes. While axiomatic approaches are clear and sharp on the first two counts, their success is more limited with respect to the third. In general, one can say there are valuable successes with respect to very general theoretical insights, such as the connection of spin and statistics as well as non-localizability, while the weak point is the lack of realistic models for interacting quantum field theories. Since the fundamental entities in axiomatic reformulations of QFT are algebras (of smeared field operators or of observables) instead of quantum fields, reformulating QFT in algebraic terms and in axiomatic terms are enterprises with a large factual overlap. However, only one attempt to reformulate QFT axiomatically bears the name “Algebraic Quantum Field Theory,” because it is here that the algebraic structure of the fundamental entities was first fully realized and is explicity at the centre. However, trying to do this in a strictly axiomatic way, one only gets ‘reformulations’ which are not as rich as standard QFT. As Haag (1996) concedes, the “algebraic approach […] has given us a frame and a language not a theory.”