Peano's five axioms for the sequence of natural numbers run as follows: (1) 1 is a number, (2) Any number α has a unique sequent a′ (3) Different numbers have different sequents, (4) 1 is not a sequent, and (5) The principle of mathematical induction.

Now, in a system of axioms, it seems desirable, in the first place, to avoid symbols for particular individuals and symbols for functions, and, secondly, to secure the property of complete independence. In fact, Peano's system can be replaced by an equivalent one which fulfills both requirements. In such a system, the symbol “1” and the function “sequent” will not appear, although they may be introduced afresh through definitions. Besides, the principle of mathematical induction will no more play the rôle of an axiom, because, under the second requirement, it can hardly be stated in a simple and elegant form.