A. M. Turing. On computable numbers, with an application to the Entscheidungs problcm. Proceedings of the London Mathematical Society , 2 s. vol. 42 (1936–1937), pp. 230–265.

Superposed upon a finite sequence of axioms (the "Rumpfsystem") each of which makes a certain assertion with regard to the funda mental concepts employed, appears a final axiom seemingly concerning the preceding axioms and not related to the fundamental concepts of the system. The best known such axiom-system is that of Hilbert for Euclidean geometry, with its famous "Axiom of Completeness." Whether the final axiom states that no more inclusive system of things exists which satisfies the preceding—and is therefore a "maximal" axiom—or is analogously a "minimal" axiom, such a final axiom will be called an "extremal" axiom. The authors defend the use of such axioms under suitable restrictions and when properly stated and interpreted. A fundamental concept in the study of axiomatics is the notion of isomorphism which the authors extend, by the concept of correlators which are binary relations between given n-ary relations. Complete isomorphism is discussed with respect to types of like speci fied order. If any two structures satisfying the "Rumpfsystem" are completely isomorphic as to elements of specified order, one may then inquire as to whether such a structure does or does not have a proper substructure isomorphic with it. Distinction is made between extensions of model and ex tension of structure. The legitimate introduction of the extremal axiom corresponds to the selection of extremal structures. The question of independence of the axioms in the "Rumpfsystem" as affected by the introduction of an extremal postulate is discussed and various cases are found to occur. A final serious question arises with regard to extension to a system of different order-type, as occurs from the system of rational numbers to that of real numbers regarded as sequences of rationals. Tarski's restriction to an increase of one unit in order type has many attractive features, and avoids certain serious difficulties, but is found to be somewhat too restrictive.